"It is incumbent on us to ensure that we have that," she said.
In an unbelievable turnaround, the odds on Corbyn becoming the next PM have been slashed, while May is already being savaged by her own party.
Voters appear to be punishing her for calling the ballot unnecessarily and a catastrophic campaign in which she was seen as having performed dreadfully.
In a speech after being re-elected in Islington North, Corbyn said May had wanted a mandate but been given a devastating verdict by voters.
"'That is enough to go, actually, and make way for another government," he demanded.
"Politics is not going back into the box where it was before."
Deputy Labour leader Tom Watson said May would "never recover" and mocked her claim to be a "bloody difficult woman", saying she was now finding it "all too bloody difficult".
Former chancellor George Osborne snarled that the numbers looked "catastrophic" for the Conservative leader.
Despite Tory hopes they would be the biggest winners from a massive Ukip collapse, the exit poll at 10pm last night indicated they were on track to lose 16 seats, leaving them on 314 and well short of a majority - while Labour was up from 232 to 266.
However, things became even more grim for May, with Education Secretary Justine Greening barely clinging on in Putney and Treasury minister Jane Ellison losing Battersea amid a London meltdown.
Cabinet Office minister Ben Gummer has been defeated in Ipswich, and there has also been a miserable showing in Wales, while Home Secretary Amber Rudd could fall with a recount under way in her Hastings seat.
The only bright spot appears to be Scotland, where they are making gains from the SNP including unseating the party's Westminster leader Angus Robertson in Moray.
The BBC is currently forecasting that the Tories could end up with 322 seats to Labour's 261, just about enough for a workable majority, but many are expecting the final tally to be lower.
Meanwhile, former Lib Dem leader Nick Clegg became the biggest casualty so far as he lost Sheffield Hallam to Labour - but former business secretary Vince Cable fared better as he made a comeback in Twickenham.
A hung Parliament could throw Britain into chaos, leaving us effectively rudderless barely a week before negotiations with the EU are due to get under way. Former Ukip leader Nigel Farage said he was ready to return to politics, warning that he feared a second referendum on Brexit.
Senior Conservatives said that May had made "fundamental strategic errors" and said that her closest aides should be "banished" from Downing Street.
They complained that the campaign had been centred around a "cult of personality" and "central control", adding: "It has completely blown up in our face".
One senior Tory told the Telegraph: "This is bad, it's worse than bad. Her advisers should walk out of the door now never to return, regardless of the final result. They should be banished forever.
"Can she hang on? She [Theresa] has zero credibility if this exit poll is correct.
"The very best we can get tonight is to end up where we were."
The survey predicted the Conservatives would get 314 seats and the Labour Party 266.
It projected 34 for the Scottish National Party and 14 for the Liberal Democrats.
The pound lost more than 2 cents against the dollar within seconds of the exit poll announcement, before recovering slightly. Based on interviews with voters leaving polling stations across the country, the poll is conducted for a consortium of UK broadcasters and regarded as a reliable, though not exact, indicator of the likely result.
If confirmed, the result will be humiliating for May, who called a snap election in the hope of increasing her majority and strengthening Britain's hand in exit talks with the European Union.
If she has failed, she could face pressure to resign.
"If the poll is anything like accurate, this is completely catastrophic for the Conservatives and for Theresa May," former Conservative Treasury chief George Osborne told ITV.
"Clearly if she's got a worse result than two years ago and is almost unable to form a government, then she, I doubt, will survive in the long term as Conservative Party leader."
The Labour campaign in Britain's snap general election "has changed politics for the better", Labour leader Jeremy Corbyn says.
"Whatever the final result, our positive campaign has changed politics for the better," Corbyn said in a message via Twitter.
Labour's manifesto had given a "radical vision for a fairer Britain", he said after an exit poll suggested Labour had performed better than expected.
"Our team has worked so hard on this campaign - from door knocking to social media - and it's great that we've won so much support across the country."
A party needs to win 326 seats in the 650-seat House of Commons to form a majority government.
The Conservatives held 330 seats in the last Parliament, compared with 229 for Labour, 54 for the Scottish National Party and nine for the Lib Dems.
During the last election, in 2015, the Conservatives did better than the exit poll predicted, and senior Conservatives said they would take a wait-and-see approach.
"It's still very, very early in the evening," Northern Ireland Secretary James Brokenshire said.
"It's too early in the night to be drawing conclusions".
The forecast is much better than expected for the opposition Labour Party, which had been expected to lose seats.
Labour economy spokesman John McDonnell, one of Corbyn's main lieutenants, urged caution.
"I'm a natural pessimist, and we'll see whether I'm an optimist in the morning," he said.
The result could also be bad news for the Scottish National Party, which is predicted to lose 20 of its 54 seats - though the pollsters cautioned that there is a lot of uncertainty around the Scottish forecast.
A big loss could complicate the SNP's plans to push for a new referendum on Scottish independence as Britain prepares to leave the EU.
Results will come in overnight after an unsettled election held in the shadow of three deadly attacks in as many months. May called the election seven weeks ago - three years ahead of schedule and with her party well ahead in the polls.
She argued that increasing the Conservative majority in Parliament would strengthen Britain's hand in Brexit talks. But things didn't go to plan.
Brexit failed to emerge as a major issue in the campaign, as both the Conservatives and Labour said they would respect voters' wishes and go through with the divorce.
Then attacks that killed 30 people in Manchester and London twice brought the campaign to a halt, sent a wave of anxiety through Britain and forced May to defend the Government's record on fighting terrorism.
Eight people were killed near London Bridge on Saturday when three men drove a van into pedestrians and then stabbed revellers in an area filled with bars and restaurants.
Two weeks earlier, a suicide bomber killed 22 people as they were leaving a concert in Manchester.
Before the election, five people died during a vehicle and knife attack near Parliament on March 22.
Rachel Sheard, who cast her vote near the site of the London Bridge attack, said the election hadn't gone as expected - and that it certainly wasn't about Brexit.
"I don't think that's in the hearts and minds of Londoners at the minute, [not] nearly as much as security is," said Sheard, 22.
"It was very scary on Saturday."
May, who went into the election with a reputation for quiet competence, was criticised for a lacklustre campaigning style and for a plan to force elderly people to pay more for their care, a proposal her opponents dubbed the "dementia tax".
As the polls suggested a tightening race, pollsters spoke less often of a landslide and raised the possibility that May's majority would be eroded.
In her final message to voters, May tried to put the focus back on Brexit.
"I can only build that better country and get the right deal in Brussels with the support of the British people," she said.
"So whoever you have voted for in the past, if that is the future you want then vote Conservative today and we can all go forward together."
Corbyn, an old school left-winger widely written off at the start of the campaign, has drawn thousands of people to upbeat rallies and energised young voters with his plans to boost spending on health and education after years of Conservative austerity.
Corbyn told supporters at his final rally that Labour's campaign had "changed the debate and given people hope. Hope that it doesn't have to be like this; that inequality can be tackled; that austerity can be ended; that you can stand up to the elites and the cynics. This is the new centre ground."
Security dominated the late stages of the campaign, after the attacks in Manchester and London. May said this week that she would consider rewriting human rights legislation if it gets in the way of tackling extremism.
-AP
What do the polls and pollsters say?
ICM predicts the Conservatives will win by 12 per cent and gain a 96-seat majority. ComRes says it will be 10 per cent and 74 seats. From eight pollsters, the Tories have leads of between 1 to 13 per cent.
Of six polls yesterday, two showed the Conservatives widening their lead over Labour, two showed a tightening and two were unchanged.
The Britain Elects twitter site predicts the Conservatives will win 356 seats, and Labour 219. It says 144 seats are at risk of changing hands.
A hung Parliament is a possibility should polls predicting a 1 to 4-point margin prove accurate. Pollsters seeing a tighter contest believe there will be large numbers of young voters casting votes for Labour.
What happened at the last election?
May currently has a working majority, with allies, of 17.
At the last election in 2015 the Tories won 36.9 per cent and 330 seats, and Labour 30.4 per cent and 232 seats.
What happened in the latest campaign?
A 10-point Conservative margin would be about half the poll gap between the Tories and Labour when the PM called the election in April.
The campaign has shown an enthusiasm gap between the main parties with Labour's Jeremy Corbyn drawing some large rallies and May far more risk-averse. Labour will be reliant on a big turnout among young voters while the Conservatives will depend on older voters, who, based on past patterns, are more likely to actually get to polling stations to vote.
Media speculation is that Corbyn - having exceeded low expectations - could emerge with his standing enhanced, whereas a squeaky scrape-through could wound May and make her vulnerable to a future challenge.
Perhaps the closeness means the electorate is showing disquiet at one-party dominance. But, just two years after the last election and with apparently no chance of Brexit being overturned, a full handover of power from the Conservatives to Labour would be a major shock.
Is the European political scene still unpredictable?
Since the body blows of Brexit and Donald Trump in the US last year, there's been less political drama than expected in Europe.
Italy changed its Prime Minister. The Netherlands leader was returned and saw off the challenge of a far-right party.
France tweaked the cut of its leader's coat, choosing an independent over the traditional parties' candidates. It still ultimately went with a centrist over a far-right challenger.
In Germany, which holds elections in September, Angela Merkel is purring away from her chief rival and towards her fourth term like a Merc in peak condition.
Stability has crept back into a political system that had previously been buffeted by populism. The US President is for many people a walking cautionary tale of inserting a populist with no political background into power.
The Daily Telegraph's election day guide
10am
Safe Labour seat Houghton and Sunderland South is expected to declare its results first so all eyes will be on the northeast. It will give us the first steer as to how Labour has done - but we can expect Bridget Phillipson to retain her strong lead.
10:30am - 11
The Tories are hoping to hoover up votes in the north of England with their stance on Brexit. Some polls have shown the Conservatives are now level with Labour in some constituencies and lead among working class voters. Labour has huge majorities in Sunderland Central and Washington & Sunderland West so they're pretty safe bets for Jeremy Corbyn.
11:30am
The result in Kettering should provide an indication of just how many Ukip voters are switching to the Tories. The incumbent Tory Philip Hollobone took the unprecedented step of signing an election pact with the party. Ukip agreed not to stand a candidate against Hollobone. The Ukip candidate won almost 8000 votes in 2015 so a strong performance for Hollobone will be welcomed by Tory HQ as evidence of a Ukip swing.
12pm
First marginal seat of Nuneaton, in Warwickshire is expected to declare around now. David Cameron said he knew the Conservatives had won when the result came in for this bellwether seat in 2015. We'll also see the first London seats come back, including Conservative stronghold Putney. The first results in Newcastle will show whether Prime Minister Theresa May's tactic of repeatedly visiting the region to woo voters has paid off. It's Labour's heartland, but strong showings for the Tories could well spell doom for Corbyn's chances of winning the keys to Number 10.
12.30pm
The Conservatives have realistic hopes of winning Darlington which is currently held by the Labour shadow Brexit minister Jenny Chapman. A defeat for Chapman would be a big loss for Labour - the party has held the seat since 1992. Rosena Allin-Khan will be hoping to win Tooting, a south London constituency she won in 2016 in a byelection after Sadiq Khan resigned to become mayor.
1pm
Three-way marginal constituency Thurrock was won by the Conservatives in 2015 by 974 votes over Labour, with Ukip just 438 votes behind. Home Secretary Amber Rudd will find out if she's kept her Conservative majority in Hastings & Rye in East Sussex. Rudd secured a majority of almost 5000 in 2015 making a loss unlikely. But if she's ousted it would represent a complete crisis for the Tories.
1.30pm
It's a big moment for Corbyn as his ultra-safe constituency Islington North declares its results. The seat has been held by Labour since 1937, and by Corbyn since 1983.
More interestingly, we'll find out if Labour has retained its most marginal seat in Wales, Ynys Mon, formerly known as Anglesey, against the confident Tories. A result should also come in for the Leigh constituency in Greater Manchester which Andy Burnham represented since 2001 before stepping down to be Mayor of Greater Manchester.
2pm
Could the Tory surge in Scotland reach as high as Angus Robertson? Toppling the SNP's deputy leader in Moray would be a major scalp and mean the Conservatives could bag 10 or more seats, but the bookies think it will hold. If the Tory surge in Scotland is genuine, they will take Pete Wishart's seat of Perth and North Perthshire from the SNP. Conservative gains would fuel the argument that support for Scottish independence is waning; failure to break through, especially here, would boost the SNP. The Tories have a "take-out Tim" strategy in Lib Dem-held Westmorland and Lonsdale, to oust Lib Dem leader Tim. His aides dismiss the idea as "nonsense".
2:30pm
Will incumbent Simon Danczuk, who is standing as an independent, beat the Labour Party in Rochdale? The MP was suspended from Labour in 2015 following allegations he sent sexually explicit text messages to a girl aged 17, so it will be interesting to see if voters back him, or the party. The Tories believe they can make historic gains in the northeast thanks to Brexit. If they take Tony Blair's old seat of Labour-held Sedgefield, then half a dozen could fall their way in the region.
3pm
All eyes will be on Gower, one of the tightest races in the UK. It's a top target for Team Corbyn which hopes to take advantage of the party's poll surge in Wales. Failure to do so, perhaps as a result of Ukip's plummeting vote share, would be a blow. Labour-held Barrow & Furness was the epicentre of the Labour moderate rebellion. Edinburgh South's Ian Murray - Labour's only Scottish MP in 2015 - is in a three-way fight to keep his seat with the SNP and the Tories pushing hard. Lib Dem-held Richmond Park is Zac Goldsmith's chance of redemption. The Tory golden boy was toppled in a byelection last December just months after losing the London mayoral race amid accusations of dirty tactics. Winning back this seat would restart his political career. Croydon Central is another of Labour's top targets and will be worth watching. Taking the seat would end a hat-trick of Tory wins and topple Gavin Barwell, the housing minister. Polls putting Labour at 17 points ahead in the capital suggest it's doable.
3:30pm
May's ultra-safe Tory constituency is Maidenhead, Berkshire, where she'll be to see the results come in. Former Lib Dem leader and deputy PM Nick Clegg will be hoping to hang on to his Sheffield Hallam seat which was heavily targeted by Labour in 2015.
4pm
Ukip's best chance of a seat will be in Boston and Skegness in Lincolnshire which is expected to declare around now. Boston was the country's most Brexit town, with 75.6 per cent voting Leave, while the Tory incumbent was a Remainer. If Paul Nuttall, the Ukip leader, fails to win then calls to resign could follow. The Tories are hoping to snatch Chester back from Labour's Chris Matheson who won in 2015 by just 93 votes. Then there's Cambridge which is a top target seat for the Lib Dems. If the party cannot take it back from Labour, a pro-EU town lost by a whisker in 2015, they face a disastrous night. The Tories have high hopes for Wales as it backed Brexit. May was the first Tory leader to visit the Bridgend constituency during the campaign in 20 years.
4:30pm
A key seat for Labour, Southampton Test, is expected to be announced about now. It is one of only a handful of marginal seats visited by Corbyn during the campaign and the party will be hopeful of holding on. A loss in Southampton for Labour would spell disaster for Corbyn's hopes of victory. Another key seat is Leeds North West. The loss of a Lib Dem seat would be a crushing blow to Farron's hopes of a resurgence for the party.
5pm
By this point the overall result should be becoming clear.
