But the motion was blocked by the DUP using the petition of concern mechanism, which means a motion must secure a majority of both unionists and nationalists.
Last night, the party was accused of "abusing power", with calls for Assembly procedures to be changed after the next election.
Demands are also increasing for a future free vote on the issue, with parties using three-line whips rather than the controversial petition, which was designed to protect minority interests.
But there were also suggestions yesterday that supporters should bring forward a Private Members' Bill rather than calling on the Executive to introduce legislation.
It was the fifth time the issue had been debated at Stormont - Northern Ireland being the only part of the UK which has not passed same-sex marriage laws - but the first time it won a simple majority.
The debate was a more mild-mannered repeat of previous exchanges but, before it got under way, two gay couples outside Parliament Buildings were handing out invitations for weddings they hope to have at some point in the future.
Only four unionists voted in favour of the measure - NI21 leader Basil McCrea, independents John McCallister and Claire Sugden, and Ulster Unionist Andy Allen.
East Belfast MLA Mr Allen, repeating the stance of his predecessor Michael Copeland, said: "I believe in equality and that love is love, whether between a man and a woman, two men or two women. In life, I have always tried to do what I believe to be right."
But other senior unionists insisted same-sex marriage was neither an equality nor a human rights issue.
Finance Minister Arlene Foster, whose department would be responsible for bringing forward legislation, argued: "This is not an equality issue. Contrary to what has been suggested, the law in Northern Ireland does not disparage or denigrate same-sex relationships, and same-sex couples are not precluded from having a loving, secure, stable and permanent relationship.
"Specific provision has been made to ensure legal recognition for same-sex relationships, and nothing - nothing - turns on the fact that same-sex couples are referred to as 'partners' rather than 'spouses'."
TUV leader Jim Allister said: "This is not a rights issue (as) recognised by the European Convention on Human Rights or the European Court of Human Rights. It therefore cannot be - and is not - a rights issue, nor is it an issue of equality."
But Alliance's Trevor Lunn, who has both voted against and abstained in the past, yesterday supported the motion and told MLAs: "I took a while to come to terms with civil partnerships away back when. The journey has been long for me, but I now am where I am, and I am satisfied and comfortable with the position that I am taking."
His colleague, Kieran McCarthy from Strangford, defied party policy by voting against, while Ulster Unionist leader Mike Nesbitt abstained.
Sinn Fein and the SDLP still hailed the vote as a breakthrough, however. Sinn Fein's Caitriona Ruane said it sent a powerful message around the world, just as the Irish Republic had done in its referendum a few months ago.
And the SDLP's Colum Eastwood said afterwards: "For too long Stormont has been about the politics of division, exclusion and hate. Today we broke that mould a little."
John O'Doherty, director of the Rainbow Project, the biggest LGBT support group in the province, said: "Today's majority vote marks another landmark victory in our campaign, and we celebrate with our LGBT friends, our families and our supporters on this momentous day.
"Our campaign continues and will not end until marriage equality is a reality for everyone in Northern Ireland."
Clare Moore, equality officer with the NI committee of the Irish Congress of Trade Unions, said that while disappointed with the DUP using a petition of concern, "we nevertheless see this as a great step forward in the campaign for civil marriage equality."