The 9.3 million scheme, which includes the front of the building being used as retail space, was given the green light at yesterday's development management sub-committee meeting - despite calls from some members for further discussion.
Labour councillor Eric Milligan suggested that the matter should be discussed at a future planning meeting, but the majority of committee members voted in favour of pushing the plan through.
Councillor Milligan said: "For over 200 years the Assembly Rooms has been there as a mark of civic pride to the city, and we are now being asked to approve a significant change in it where a commercial dimension is brought in at ground floor level, and the planning committee are not even given the opportunity to reflect upon that. I think it's wholly unacceptable."
However, convener of the planning committee, Councillor Jim Lowrie, said committee members had the chance to request that a presentation on the plan was made by council officials at the meeting.
He said: "The Assembly Rooms application was not selected for presentation to the committee and no request was made to arrange for it to be presented. All councillors are aware of the established procedures - if anyone had wanted to ensure that the application was presented, they should have submitted their request accordingly. These claims are nothing more than mischief-making."
Almost 4,800 people have backed an 11th-hour campaign, via a petition on the Save the Assembly Rooms website, which is aimed at halting the revamp of the Fringe venue.
The historic building is set to close for at least 18 months by the end of the year, ruling it out of use for next year's Fringe. The project will also see the loss of five theatre spaces.
The programme of work will see the downstairs turned into shop spaces at the front of the building on George Street, and a restaurant at the back with an entrance on Rose Street.
Meanwhile, the artistic director of Assembly Theatre, William Burdett-Coutts, who is opposed to the refurbishment plans, was also astonished at the decision.
He said: "It's a grand old historic building, and I don't see that much wrong with it. If it's not broken, why fix it?
"I can't believe there hasn't been any discussion about the plans - it just seems extraordinary. These plans have been dreamed up by a few in-house people at the council who have ignored the rest of the world."