"Thank you for this extraordinary honour," Albanese said, adding Australians had voted for change.
He was humbled and honoured to serve as the 31st Prime Minister of Australia, he said, fighting back tears.
"My Labor team will work everyday to bring Australians together and I will lead a government worthy of the people of Australia," he said.
"A government as courageous and hard working and caring as the Australian people are themselves."
He spoke about his life as the son of a single mother growing up in a council flat in inner-city Sydney.
He wanted to bring Australians together, he said.
"I want to seek our common purpose and promote unity and optimism, not fear and division," he said.
"I want to unite people.
"I think people have had enough of division. What they want is to come together as a nation, and I intend to lead that."
He thanked "each and every one" of the "true believers of the Australian Labour Party" and the "mighty trade union movement".
He also thanked his "proudest achievement" - his son Nathan - and his girlfriend Jodi Haydon.
He wished outgoing PM Scott Morrison and his family well and thanked him for his service.
"Scott very graciously wished me well. And I thanked him for that and I wish him well," he said.
"And I thank him for the service that he has given to our country as Prime Minister."
Albanese also acknowledged Jenny Morrison and Morrison's two daughters for their "contribution and sacrifice".
Morrison conceded the 2022 election just before 1am (NZ time) and also said he would step down as leader of the party.
Speaking from the Liberal Party HQ in Sydney, Morrison wished the incoming Prime Minister the best of luck moving forward.
"Tonight, I have spoken to the Leader of the Opposition and the incoming Prime Minister, Anthony Albanese. And I've congratulated him on his election victory this evening," he said.
"On a night like tonight it is proper to acknowledge the functioning of our democracy. I have always believed in Australians and their judgement and I've always been prepared to accept their verdicts and tonight they have delivered their verdict and I congratulate Anthony Albanese and the Labor Party and I wish him and his government all the very best," he said.
"Now there are many votes still to count, that is true. There are many pre-polls and postles that will still come in. But I believe it's very important that this country has certainty. I think it's very important this country can move forward."
Morrison also confirmed he would be resigning as the leader of the Liberal Party.
"To my colleagues tonight, who have had to deal with very difficult news, and have lost their seats tonight, I as leader take responsibility for the wins and the losses. That is the burden and that is the responsibility of leadership," he said.
"As a result I will be handing over the leadership at the next party room meeting to ensure the party can be taken forward under new leadership which is the appropriate thing to do."
At 1am today NZ time, the ALP had won 69 seats, the Coalition 50, Independents 10, Greens two and others two. More than two dozen seats remained in doubt. Seventy-six seats are needed to govern as a majority.
Earlier report
Nine News political editor Chris Uhlmann last night said there is "no way" Scott Morrison could form government.
"It'll be a matter of what kind of government does Anthony Albanese form. Does he get to 76 and beyond and have a majority government or does have we have quite a large and unwieldy crossbench," he said.
He predicted that once the two-party preferred votes are distributed then the "teals" will help blow Liberals out of the inner-city seats in Victoria.
Uhlmann said it was looking more and more likely that Morrison will no longer be the leader of the country.
"So Scott Morrison will cease to be the ... Prime Minister of Australia. The only person who can be is the Anthony Albanese, so we will see what happens next," he said.
Uhlmann said there is "not enough numbers in the parliament" for Morrison to be able to negotiate.
"The only person in position to form a government is Anthony Albanese. It looks like the Labor Party will get there as a majority government," he said.
ABC analyst Anthony Green also claimed that it is unlikely Morrison will be able to form government, however, that didn't mean Labor would be able to form a majority.
"It is not clear that Labor will reach majority. It is clear the Coalition won't reach majority and I'm not sure what they can cobble together to try to govern," Green said.
"I would say the Coalition Government has been defeated but at this stage it is not clear what sort of Labor government we will get. There is more counting to come."
Seven News political editor Mark Riley also made the call that Morrison was out.
"We have made the call. Scott Morrison is no longer prime minister, he is in minority," he said.
And commentator Peter van Onselen said he didn't think the Coalition could win the election.
The Australian's contributing editor said about 10.30pm (NZ time) that "Scott Morrison has lost this election".
"The Liberal Party cannot form government, in my opinion," van Onselen said.
"We are now going to get into the entrails of whether that is a minority or a majority," he said.
The University of Western Australia professor of politics said the Liberals were losing too many seats — partly because of the so-called "teal" independents backed by Climate 200 — for them to be competitive to form a minority government.
"That is not going to happen," he said.
Van Onselen said Labor was looking good to potentially get a narrow majority and even if they didn't, the minority government they would be able to form was "clear to see" because the Greens and independent Andrew Wilkie would back them in.
He also noted Warringah independent Zali Steggall said she would not back a Coalition government under Scott Morrison.
"Scott Morrison has lost this election," van Onselen said.
"What we are waiting to see is by how much at this point in time."
Van Onselen described the result as a "war on two fronts" with the Coalition battling the teal candidates as well as Labor.
"They were unable to pull it together," he said.
"Labor can lock in the gains even if they are in a minority government."
"Anthony Albanese will be the 31st Prime Minister of Australia and these are the guys who got him in," Nine political reporter Chris O'Keefe said, referring to the room full of people at Labor HQ.
"Volunteers. Young Labor supporters. Albo is going to be the Prime Minister. He is a local lad. Grew up in the area. He's from the area. This is NSW and Australia Labor."
Labor's Jason Clare claimed "the teals are eating the Liberal Party alive".
How the night unfolded
