Council members voted in favour of the scheme, which will now be referred to London mayor Boris Johnson for approval.
"We hope to get a decision from the mayor by the start of December," said Real Estate Opportunities planning director Jeremy Castle. "If he is happy with it and it doesn't get referred to the secretary of state, the next stage will be to finalise the section 106 agreement and crack on with reserved matters and the first phase of the project."
Overall the redevelopment will see the construction of more than 3,400 homes across the 16ha site. A pedestrian high street and square are part of the masterplan, with space set aside for shops, restaurants, bars, cafes, offices, a hotel, a cinema and community uses.
A new section of the Thames Path is to be created along with a riverside park in front of the power station. Work to repair the landmark is scheduled at an early stage of the project and it is planned that the building will be structurally secure by the end of 2016.
The approval includes permission for the power station's chimneys to be rebuilt, a move condemned by former Wandsworth conservation officer Jack Warshaw.
"I am concerned that the council hasn't secured a guarantee that the chimneys will be rebuilt if the developer goes bust or the site gets sold," he said.