The planning committee meeting heard concerns about the potential loss of views, with heritage groups worried the sight of St Paul's Cathedral will be lost for some.
Westminster councillors voted 3 – 1 for the bridge. Lambeth council approved planning permission last month.
The £175m footbridge across the River Thames, which has been called the most expensive in the world, has been widely welcomed by horticulture and landscape professionals but the issue of who will foot the £3.5 million yearly maintenance bill remains contentious. Both planning authorities have put a condition on their permission that Transport for London must act as a guarantor of the maintenance costs.
Hetherwick’s design, with hollow legs to allow for growing media and roots, has been realised by Arup, which is doing hard landscaping as well as civil engineering. Dan Pearson is responsible for soft landscaping and the planting design.
Bridge consultant Simon Bourne told BBC London the cost is "five to 10 times more than you'd expect a footbridge to be" because of the its copper cladding.
He said: "One has to view this bridge to a certain extent not as a piece of infrastructure, but as a piece of art. It's not value for money and I don't think it's good design."
Transport for London has agreed to pay £30m in enabling costs to be matched by the Treasury. The Garden Bridge Trust says it has raised about £110m which will go towards running costs.
Trust chairman Lord Mervyn Davies said: "I am delighted with the news that Westminster City Council has shown its support for the Garden Bridge and approved our plans. This iconic new project will create a new green space in the centre of London and will be enjoyed by all."
"The bridge was the brainchild of actress Joanna Lumley. Pearson told Horticulture Week that she was "immensely enthusiastic and hugely articulate and she has an infectious desire for it be something that becomes real."
He said the bridge will provide both jobs and apprenticeships where Londoners can learn important horticultural skills.
The trust has dismissed reports of ticketing entrance to the bridge as nonsense. "It will be free and open to all" a representative said today.
The Mayor of London will make the final decision.