Scottish convener Lilian Macer said the body representing public service workers had backed calls for another vote, at a time to be decided by the Scottish Parliament.
Speaking after a meeting in Glasgow, Ms Macer said: "This in no way pre-determines the views our members may take in the event of an independence referendum, but they should have the opportunity to express their views.
"Unison Scotland defends public services and those who deliver them and it is imperative that we explore the full range of options available to the people of Scotland."
The motion put to members stated that Unison "supports the call for a second referendum, at a time to be determined by the Scottish Parliament, by means of either a section 30 order or an amendment to the Scotland Act as a satisfactory means of transferring the power over independence referendums."
On Thursday MSPs voted in favour of a motion calling for a second referendum.
The Scottish Greens welcomed Unison's endorsement, describing it as a "totemic moment" for the independence movement.
Co-leader Patrick Harvie MSP said: "Unison, Scotland's largest trade union, represents members of different party political persuasions, but its decision today to back an independence referendum, at a time to be determined by the Scottish Parliament, recognises the democratic deficit that we currently face.
"This can only be resolved by putting the question of Scotland's future back to the people."