Construction will begin in spring 2015 and it is hoped the line, which will be extended from Kennington to Battersea, will open by 2020.
Two new stations will open - Nine Elms to the east and one at the heart of the Battersea Power Station development.
The full cost is expected to be up to Â£1bn, which will be funded by developments in the area.
Ravi Govindia, leader of Wandsworth Council, said: "This is fantastic news for the people of Battersea and the benefits will be felt right across Wandsworth.
"This part of the borough is already coming to life with thousands of new homes and jobs and its two new Tube stations will turbo charge the process.
"The Northern line extension will take the pressure off the rest of our local transport network and gives Nine Elms official Zone One status."
The project was approved by the Secretary of State for Transport.
The decision follows a public inquiry conducted by an independent inspector last year to consider the impact of the extension.
Some residents had opposed the plans, saying it would lead to a loss of parkland and increase noise and pollution in the residential area.
The London Underground's Northern Line extension to Battersea has been given the go-ahead.