As vote counting continued, at 9.30pm Labor was leading in 73 seats, to the Liberal-National coalition's 54.
"We expect to be in a position to form government and hope to be in a position to form majority government," a spokesman told AAP on Saturday night.
The crossbench will be at least 13-strong, with the remaining 11 seats in doubt.
With 40 per cent of the vote counted, the coalition was on 35 per cent of the primary vote to Labor's 31.6 per cent.
The Greens were sitting on 12.6 per cent of the primary vote, while independents held just under six per cent.
The most likely outcome on current trends is a Labor minority government.
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.
Treasurer Josh Frydenberg appeared on track to lose his seat of Kooyong but was not conceding.
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.
Labor frontbencher Jim Chalmers said a majority Labor government was "still possible", but he did not think the coalition could retain power.
"There are a lot of results where the second place getter and the third place getter is not absolutely determined," he told the ABC.
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 appeared on track to hold 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."
Former Liberal minister Christopher Pyne earlier predicted a Labor win, expecting the party will gain seats numbering in the "high 70s".
"If we win today, it will be very surprising," Mr Pyne told the Seven Network.
"Labor will win but, I think things very much tightened up in the last six weeks and what looked like a blowout … is obviously much closer."
Both major party leaders will be attending functions in Sydney on Saturday night.
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