The League Two club applied to build an 11,000-seater stadium - that could be expanded to hold 20,000 - on the site of Wimbledon greyhound stadium.
Merton Borough Council had recommended it for approval and councillors unanimously approved the plans.
The original Wimbledon club left Plough Lane in 1991 because of legislation requiring all-seater stadiums.
Wimbledon FC, as it was known then, shared Crystal Palace's Selhurst Park until 2003 when it was given permission to relocate about 70 miles (112km) north to Milton Keynes and was later renamed MK Dons.
Many fans protested against the move and decided to form AFC Wimbledon as a non-League club.
The side rose back to league status and currently plays at Kingsmeadow in Kingston upon Thames,
The AFC Wimbledon Supporters Trust has agreed to sell its lease at Kingsmeadow to Premier League Chelsea to help fund the new stadium.
The Plough Lane planning application that has been approved also includes 602 residential units, along with retail and commercial space and a leisure club.
AFC Wimbledon's chief executive Erik Samuelson said in a message on the supporters' trust website: "This is an important step in a journey we started some years ago.
"We have a long way to go and many major hurdles to clear but we want to build a stadium that is embedded in the community and operate it in an inclusive way."
The Greyhound Board of Great Britain, which grants greyhound racing licenses, said it was worried about the future of the sport in London if the Wimbledon track goes.
A spokesman said: "Wimbledon is the last track within the M25 and we think it is important there is a track here in London, because greyhound racing is still the fifth most popular sport in this country."
AFC Wimbledon has been given permission to build a new stadium close to its spiritual home in Plough Lane.