The new plans will bring eurozone monetary policymakers closer in line with the US Federal Reserve and Bank of England, which have both already raised rates multiple times this year.
It could leave the central banks of Japan and Switzerland as the last two major monetary authorities still deploying negative rates.
Lagarde, the ECB president, and chief economist Philip Lane had previously said rate rises of a quarter of a percentage point were the "benchmark" for its meetings in July and September.
However Lagarde stressed on Thursday afternoon that risks to the inflation outlook were now "primarily on the upside". Officials at the central bank are increasingly concerned that higher wages and continued disruption to global energy markets and supply chains will lead to inflation becoming entrenched.
The decision to pre-announce the July rise and the likely 50 basis point move in September was unanimous, with doves on the governing council avoiding a half percentage point rise next month at the cost of agreeing to open the door for an increase of that size at the following meeting.
"A lot of the tools we have were about bringing inflation up, but now we are in the opposite situation and we need to bring it back down," Lagarde said, adding that the ECB would "stay the course and be determined".
The bank last raised rates in 2011 and its deposit rate now stands at minus 0.5 per cent after hitting zero in July 2012 and diving into negative territory in 2014, during a regional debt crisis.
Governments' borrowing costs rose in response to the hawkish shift. Germany's 10-year bond yield climbed 0.09 percentage points to 1.45 per cent. Riskier debt sold off more sharply, with Italy's 10-year yield up 0.25 percentage points to 3.62 per cent.
As planned, the bank announced it would end its remaining €20bn-per-month bond purchases at the start of July.
The ECB said in a statement that its governing council "intends to raise the key ECB interest rates by 25 basis points at its July monetary policy meeting". It added that, if the inflation outlook persists or deteriorates, "a larger increment will be appropriate at the September meeting".
Beyond September, the ECB said it "anticipates that a gradual but sustained path of further increases in interest rates will be appropriate".
The sell-off in bond markets, however, underlined how the ECB's plans to ditch its crisis-era stimulus could cause problems for countries with higher debt burdens, such as Italy.
Former ECB president and current Italian prime minister Mario Draghi said on Thursday ahead of the meeting that rising inflation in the EU was "not wholly the sign of overheating but largely the result of a series of supply shocks". There was, he said, "still spare capacity in the economy".
Investors, meanwhile, were keen for the ECB to explain what it planned to do to avoid the risk of a repeat of the financial market stress that led to bailouts of several countries, including Greece, Portugal and Ireland, during Europe's debt crisis of 2012 that nearly destroyed the single currency.
Lagarde said the ECB had to ensure the end of holding ever-increasing amounts of government debt and higher rates did not lead to "fragmentation" of what it cost governments and businesses in individual member states to borrow. "We know how to design and we know how to deploy new instruments, if and when necessary," she added. "We have demonstrated that in the past; we will do so again."
The ECB president said the bank had full flexibility to decide when and how to deploy its "anti-fragmentation" tools, including the ability to reinvest proceeds of maturing securities in a €1.7tn bond-buying scheme launched during the coronavirus pandemic, adding: "We will prevent it."
But she rebuffed requests to give more detail on the circumstances that could lead to it launching a new bond-buying tool, saying: "There is no specific level of yields increase, or lending rates or bond spreads that can unconditionally trigger this or that."
The gap between Italian and German 10-year borrowing costs widened to 2.17 percentage points — the most since the early stages of the pandemic.
"We almost see the hawks taking over a little bit at the ECB," said Katharina Utermöhl, senior Europe economist at Allianz. "If there was a compromise it was one to rein in the hawks rather than to please the doves."
Italian bonds were being hit by an "unpalatable combination" of the prospect of chunkier rate rises along with a lack of detail on any new tools to keep a lid on sovereign bond yields, said Richard McGuire, a strategist at Rabobank. "As we do not know where the pain point is, this is absolutely an invitation for pressure to build."
Lagarde said on Thursday that inflation would "remain undesirably elevated for a period of time" after an "unprecedented" three-quarters of the items used to measure inflation increased in price by more than 2 per cent last month.
The pace at which price pressures have intensified over recent months has left hawks worrying that the ECB is behind the curve on tightening policy and calling for more aggressive moves, in line with the Fed's strategy of raising rates by 50 basis points at a time.
With Russia's invasion of Ukraine already driving up food and fuel prices for European consumers, there are also growing fears among economists that if Russian gas supplies are cut off it could plunge the eurozone into recession.
The ECB slashed its growth forecasts and raised its projections sharply for inflation. Eurozone inflation would increase from 2.6 per cent last year to 6.8 per cent this year, before declining to 3.5 per cent next year and 2.1 per cent the following year — remaining above the 2 per cent target for the entire forecast period.
In March, the central bank projected inflation of 5.1 per cent this year, tailing off to 2.1 per cent next year and 1.9 per cent in two years' time.
Growth would hit 2.8 per cent in 2022, 2.1 per cent in each of the next two years, well below its March projections, it said. Carsten Brzeski, head of macro research at ING, said the forecasts were still "far too optimistic, which will hinder the ECB from doing what it intends to do" in tackling inflation.
There has been speculation about how quickly the ECB could start shrinking its balance sheet by not reinvesting the proceeds of maturing bonds in the €4.9tn portfolio it has built up. But Lagarde said this question had not been discussed this week and would be left for a future meeting.
The bank said such reinvestments would continue "for an extended period of time past the date when it starts raising the key ECB interest rates and, in any case, for as long as necessary to maintain ample liquidity conditions and an appropriate monetary policy stance".
- Additional reporting by Amy Kazmin