As vote counting continued, at 10.30pm Labor was leading in 73 seats, to the Liberal-National coalition's 55.
The crossbench will be at least 13-strong.
Mr Morrison told supporters in Sydney on Saturday night he had spoken to Mr Albanese and congratulated him on his election win.
"In this country, at a time like this, when we look around the world, and particularly when we see those in the Ukraine fighting for their very freedom and liberty, I think on a night like tonight we can reflect on the greatness of our democracy," he said.
"It is proper to acknowledge the functioning of our democracy. I've always believed in Australians and their judgment 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."
With 40 per cent of the vote counted, the coalition was on 35 per cent of the primary vote to Labor's 32 per cent.
The Greens were sitting on 12.5 per cent of the primary vote, while independents held just under six per cent.
Labor's highest profile loss is frontbencher and former NSW premier Kristina Keneally who was seeking to shift from the Senate to the lower house seat of Fowler.
On the coalition side, Treasurer Josh Frydenberg appeared on track to lose his seat of Kooyong but was not conceding.
The man who was considered a future party leader said holding the seat was "mathematically possible", but difficult.
"I thank Scott Morrison for what he has done for our country to leave Australia in a stronger position than when he found it," he told supporters in Melbourne.
Defence Minister Peter Dutton, who appeared set to hold his Brisbane seat of Dickson, is widely expected to become opposition leader.
"We live in a wonderful country and we have many challenges ahead," he said.
Official figures showed Labor incumbents trailing in Gilmore and Lyons, while Liberals were behind in Wentworth, Chisholm, Brisbane, Mackellar, Higgins, Reid, Robertson, Ryan, Boothby, Sturt, Deakin, Pearce, Hasluck, Curtin and Swan.
Labor could lose the Brisbane seat of Griffith to the Greens.
Finance Minister Simon Birmingham said the coalition had lost the "are you in touch" question in many electorates, while Labor had "failed to win the best able to govern".
"That's why we're seeing a situation where our vote is down significantly, yet the Labor Party, who could form government out of tonight, have their lowest primary vote since 1919 at this stage."
Labor deputy leader Richard Marles said his party did not "escape judgment" at the election.
Labor frontbencher Tanya Plibersek said it appeared many people who had voted Liberal in the past had opted for independent or even Greens candidates in Sydney, Brisbane and Melbourne seats.
"They are people who have always voted Liberal in the past," she said.
"It is a big jump for someone who has always voted Liberal to make the jump to Labor.
"Those voters are trying to send a message that climate change is important to them, a national integrity commission with teeth is important to them, and equality with women."
Independent MP for Warringah, Zali Steggall, who held her Sydney seat, said she expected more community independents would be elected.
"People are really frustrated," she said.
"Communities are turning to alternatives to the major parties."
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