Early election counts show the Queensland Labor-held seats of Brisbane, Ryan and Griffith as well as Macnamara in Victoria could go to the Greens.
Former Labor leader Bill Shorten said affluent people voted Greens because they don't have to worry about their finances.
"Part of the challenge for Labor in the inner-city seats, and I have to contend with the Greens in my seat too, is very affluent people tend to vote Green because they don't have a worry in the world," he told the Nine Network.
Former Liberal foreign minister Julie Bishop agreed and called it a "luxury" to vote for the Greens.
But Mr Bandt said voters told him they were supporting the party for the first time this election not only because of climate action but because they didn't see many key policy differences between the government and the opposition.
He credited a people-powered campaign in Queensland and as polls started to close on Saturday night, Mr Bandt was confident the minor party would pick up spots in the Senate as well as some lower house, inner-city seats.
"We didn't go small target. We were very clear that there is a better and fairer way and that is a big part of the reason why we are seeing the results," he told ABC News.
If the Greens do hold the balance of power on the crossbench, they would approach the parliament with an open mind, Mr Bandt pledged.
"It's stable and effective and progressive government that would be our priority with action on climate and action on inequality," he said.
The Greens held the balance of power along with independents on the crossbench following a hung parliament at the 2010 election.
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