If the lower house votes to amend the bill, it will need to be returned to the Senate.
The bill passed its third reading with only four members of Parliament in opposition.
In a historic move, the same-sex marriage bill was passed through the House of Representatives.
"Love has won, and it's time to pop the bubbly", Greens MP and same-sex marriage supporter Adam Bandt said. More than 100 MPs had delivered speeches, many emotional, in the lead-up to the final vote with most in support of changes to the Marriage Act.
The first gay marriage is supposed to be registered in February 2018. "Australia has done it", said Mr Turnbull just before the parliamentary vote.
"Some individuals who entered into marriage in their previous sex or gender are faced with a choice between divorcing their spouse in order to obtain records reflecting their gender identity, or not having these records", the bill's explanatory memorandum notes.
The vote came after Australians voted "yes" following a two-month national same-sex marriage survey.
Among them was Christine Forster, the gay sister of Tony Abbott, the former Australian Prime Minister who has always been seen as the face of the "no" campaign.
Abbott and other conservatives proposed amendments to the bill to include further religious protections, all of which were voted down. The major parties want the legislation passed this week after a majority of Australian's endorsed change in a postal ballot last month.
The Equality Campaign said: "This Bill ensures every LGBTI Australian will now be treated equally with the same dignity and respect as their fellow Australians and will be able to marry the person they love".
About 12.7 million people participated in the survey, which had a response rate of almost 80% - which is considered very high.
Australia is the 25th country in the world to have marriage equality for same-sex couples.