"It would be like celebrating the Nazi holocaust", Socialist Alliance councillor Sue Bolton told the meeting on Wednesday.
But it is termed "Invasion Day" by many indigenous Australians who say it marks the beginning of the decline of Aboriginal culture.
Moreland Council, in Melbourne's north, voted 7-4 on Wednesday night in favour of the motion - making it the most contested vote of the three Melbourne councils to pursue the change.
A third local city council in Australia's second-largest city Melbourne has voted to abandon official Australia Day celebrations on January 26, claiming they are insensitive to Australia's Indigenous people. Councillors also voted to join the push to change the date.
"In particular, the government absolutely condemns the extreme and divisive nature of the discussion Greens and Socialist councillors are promoting", Mr Hawke said.
The Turnbull government has criticised the councils and said any comparison with the Holocaust was offensive.
"We can't have individual councils across the nation changing the date unilaterally, it's a matter for the federal government, and the federal government believes it should be on 26 January", he said.
But Greens councillor Samantha Ratnam said she was "proud of council for taking this step".
Assistant minister for immigration and border protection Alex Hawke, in a statement, said ratepayers did not sign up to have Australia Day dumped.
"The government will consider (the) Moreland motion, and the nature of the public debate and consider further action as appropriate".
However, the council stopped short of cancelling citizenship ceremonies to avoid losing its right to host them altogether. In June, a proposal to change the date of Moreland's January 26 citizenship ceremony out of respect for the Aboriginal community was voted down.
"Thousands of Australians like me want to celebrate Australia Day - just not on a day that is painful for others".
Media captionWhy is Australia Day controversial? "To take those things away from us whether we be Indigenous or non-Indigenous, it's not right".
The federal government last month stripped Yarra and Darebin councils of their right to host citizenship ceremonies after successful votes to shift them to another day.
The annual holiday, on Jan 26, commemorates the arrival of the country's first British settlers in 1788 and is a time when citizenship ceremonies are held.