The plans for 39 new grass courts, including a new 8,000-seater show court, will transform adjacent open land including a golf course and Wimbledon Park.
At a public hearing at City Hall on Friday, Jules Pipe, Deputy Mayor for Planning, Regeneration and the Fire Service, said the controversial scheme would keep the Wimbledon Championships as the world's pre-eminent tournament and would provide "significant benefits" to the local area.
Opposition to the plans that will use metropolitan open land to triple Wimbledon's footprint included local residents' groups, environmentalists and some local MPs.
Cries of "shame" were heard in the crowd when the GLA decision was announced.
Debbie Jevans, Chair of the All England Club, welcomed the news. "We are delighted that the Greater London Authority has resolved to approve our applications to transform the former Wimbledon Park Golf Course," she said in a statement.
"Our proposals will deliver 27 acres of newly accessible parkland for the community and enable us to bring the Qualifying Competition for The Championships onsite, with all of the substantial economic and employment opportunities this presents."
Opponents argued that the expansion would create a "tennis industrial complex" that would be unused for most of the year and would cause environmental damage.
"Every stage of this project will be delivered with meticulous attention to detail and the utmost respect for both our neighbours and the environment," Jevans said.
While facilities at Wimbledon are highly regarded, the qualifying competition takes place a few miles down the road at Roehampton.
The All England Club say moving the qualifying tournament onsite would be popular with players and fans with as many as 10,000 able to attend each day, rather than 2,000 at Roehampton.
It also says it would mitigate environmental impacts by improving the Wimbledon Park lake and planting 1,500 trees.
Seven of the new Championship standard grass tennis courts would be open for community use, according to AELTC.
Friday's decision could mark the end of the road for opposition who could potentially have called for a judicial review into the AELTC's plans.
However, Deputy Prime Minister Angela Rayner, the Secretary of State for Housing, Communities and Local Government, has said she is content for the GLA to rule on the application.
In a statement on Friday, Pipe said: "These plans for the site of a former private golf course will bring significant benefits to the local area, the wider capital and the UK economy, providing increased access to open green space and sport, new parkland and a host of new jobs.
"Hosting qualifying events on the same site as the Championships will put Wimbledon on a global footing with other Grand Slam tournaments and ensure it remains one of the world’s top sporting events."
While considering the benefits and disadvantages of the proposals, the GLA projected that the plans would be worth in the region of 336 million pounds ($449.53 million) to the UK economy each year and would support 40 new jobs across the year and more than 250 new jobs during the Championships.
The plans had been given the green light by the London Borough of Merton last year but the planning committee of neighbouring Wandsworth, where a small section of the AELTC's plans come under, had turned it down.
The decision was subsequently referred to the Greater London Authority. London Mayor Sadiq Khan had excused himself from the process due to his public support of the plan three years ago.
