In a report last week, planning officials recommended that the designs for the futuristic spiral be refused on the ground that the building was too large for the site.
But Kensington & Chelsea planning committee overruled them yesterday and voted eight to four in favour of granting permission.
David Campion, the planning and conservation committee chairman, said: "After the most careful consideration we have decided that on balance we should grant permission.
"Of course we are concerned about the impact on the surrounding buildings but, because of the proposal's national importance, it was felt that this transcended the borough's planning policies."
A spokesman for the V&A said last night that everyone at the museum was "ecstatic".
Dr Alan Borg, director of the museum, said: "We know there is a lot of hard work still to do but this is a marvellous vote of confidence in the quality of Daniel Libeskind's proposal."
Sir Jocelyn Stevens, the chairman of English Heritage, who has been a vociferous supporter of the design, said it would have been a tragedy if it had been refused planning permission. "A building can change the whole perception of a town. The Sydney Opera House changed Australia and this building is as important," he said recently.
The pounds 75m building, designed by the avant-garde architect Daniel Libeskind, is intended to sit between the museum's Henry Cole Wing and the Aston Webb site in south Kensington. The architect has described it as "a geometric spiral". Lifts would scale the building's exterior and it would house galleries and an education centre.
But ever since the design was unveiled two years ago it has attracted support and opposition in equal measure.
Brian Sewell, the art critic, said the building was totally inappropriate for the site and he was appalled at the decision.
"As a separate design in the middle of Milton Keynes it might have something to be said for it but it has no business being at the V&A," he said.
"The people who made this decision ... should never have been allowed to make a decision of this magnitude without having a look at some of the other work he [Libeskind], has done. But there is still a chance that it may not be built because they have yet to find the money."
Many local residents are strongly opposed to the extension. A spokesman for the Chelsea Society said the building was deliberately designed to shock, and the "Exhibition Road is not the Bilbao waterfront".
Carol Seymour-Newton, of the Knightsbridge Association, said the building lacked dignity and grace and added that members were very disappointed. "We felt it was discordant, but these things happen and perhaps we will get used to it in time."