The £175m footbridge would link Temple with the Southbank, but has been criticised over its location and cost.
Already approved by Lambeth Council, it is the idea of actress Joanna Lumley and has been designed by London 2012 cauldron creator, Thomas Heatherwick.
The Mayor of London will make the final decision.
Westminster councillors voted 3-1 in favour of the bridge which will cost £3.5m in annual maintenance.
At a planning meeting concerns were raised about the potential loss of views, with heritage groups worried the sight of St Paul's Cathedral will be lost for some.
Civil engineers have also called it "the most expensive footbridge in the world".
Bridge consultant Simon Bourne told BBC London the cost is "five to 10 times more than you'd expect a footbridge to be".
He said that is because of its copper cladding.
By Tom EdwardsTransport correspondent, London
Campaigners feared the planning committees were just a formality as both Lambeth and tonight Westminster had nothing financially to lose with a garden bridge.
In a hot, packed room on the 17th floor there were shouts of "shame" and "you're turning it into a theme park" from some in the audience.  Supporters kept quiet.
And while both councils gave planning permission to the garden bridge, both committees have attached strings to the project. Westminster said Transport for London must act as guarantor on the £3.5m annual maintenance costs or the project won't get built.
Will TfL like that? Will the mayor tell them to underwrite maintenance or could there be another solution like a surety? The Garden Bridge Trust will be delighted at gaining permission. This is a big step for them but campaigners will already be plotting other ways to stop the bridge.
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."
But Lord Mervyn Davies, chairman of the Garden Bridge Trust, said: "I think it's a reasonable cost for what is going to be an iconic bridge and I think the cladding is important because it needs to be visually beautiful.
"It needs to be able to stand the test of time."
Robert Davis, deputy leader of Westminster Council, chaired the planning committee and said: "This is something that is iconic and absolutely unique, and will be recognised right across the world.
"I understand the concerns about potential loss of views, but there is no doubting that this bridge will bring substantial and significant benefits to London."
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 £120m which will go towards running costs as well.
As the bridge is a major infrastructure project, the Mayor of London will have the option to call it in for further scrutiny, or allow the bridge to go ahead as set out in the applications that were put before Lambeth and Westminster councils.

Shares in the company more than halved in April when Imagination said that Apple was to stop using its technology.
The US company uses the UK firm's chip technology in its iPhones, iPads, and iPods under a licensing agreement.
Imagination said it had received interest from suitors in recent weeks so had begun a formal sale process.
It added it was now in preliminary talks with potential bidders.
In April, Imagination said that Apple, which accounts for about half of Imagination's revenues, was planning to stop using its technology within "15 months to two years".
At the time, it questioned whether Apple would be able to develop its own computer chip designs without breaching Imagination's patents.
Last month, Imagination announced it had started a dispute resolution procedure with Apple over licensing payments.
In its latest update, the company said it remained "in dispute" with Apple.
Separately, Imagination said the sale process for its MIPS and Ensigma businesses - which was announced in May - was "progressing well", with indicative proposals received for both units.
News of the sale sent shares in Imagination up 15% in early trade to 142p. However, the share price still remains nearly 50% below the level it was at before Imagination announced it was being dropped by Apple.
Plans to build a garden bridge across the River Thames are a step closer after Westminster City Council approved planning permission.
[NEXT_CONCEPT]
UK chip designer Imagination Technologies - which is in dispute with Apple, its largest customer - has put itself up for sale.