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.

Nearly 25,000 signatures - 10% of Cardiff's voters - are needed to force a vote on the issue.
But Labour councillor Ashley Govier told BBC Wales only about 8,000 have been collected.
Cardiff council's Labour group said it was "willing to take the lead from public opinion on this issue".
The Mayor for Cardiff campaign was launched earlier this year with a budget of more than Â£20,000.
Daran Hill, from the campaign, said at the time that an elected mayor was a "new, fresh idea" that could "energise the people of Cardiff".
There are currently 17 directly-elected mayors in England with more on the way, but there are none in Wales.
Mr Govier, a councillor for Cardiff's Grangetown ward, said the debate has "moved on" and a regional mayor was now needed.
"We're not going to hit the target, especially because the council refused to allow online petitions," he said.
"We feel we've moved the debate on and the focus now needs to switch to a regional mayor for south east Wales. We have to go bigger."
Mr Govier claimed the Cardiff Capital Region City Deal - a funding package agreed between UK, Welsh and local government worth Â£1.2bn over 20 years - was beginning to stall "because of governance".
"The region has to work together on infrastructure and transport," he said. "A regional mayor is needed."
He added there are "sympathetic ears throughout the region and across political parties".
Ceredigion is the only Welsh local authority to have had a referendum on the matter. Voters there rejected the idea in 2004 by a margin of nearly three to one.
A spokeswoman for Cardiff council's Labour group said: "Cardiff Labour are willing to take the lead from public opinion on this issue, and as we have seen there is very little interest in introducing a further level of bureaucracy.
"The City Deal is definitely not stalling and is moving forward, despite uncertainty caused by Brexit.
"It will be a huge boost for Cardiff and the region, and suggestions to the contrary are just mischief making."
A Wales Office spokesman said: "The Cardiff City Deal is the biggest of its kind in the UK and we are confident this ambitious project is on target."
The Welsh Government declined to comment in response to Mr Govier's comments.

Martin made 38 appearances last season for Livi, scoring five goals, as they finished sixth, one place above Rovers.
He was in his second spell with the West Lothian club, with whom he started his career.
Scott went on to play for Ross County and Hibernian, who farmed him back to the Dingwall club on loan.
He switched back to Livingston last summer.

Several fire appliances were called to the scene at Carnsalloch House to tackle the blaze.
One witness reported damage to the main part of the house at Kirkton, with floors and the roof burnt out.
The category A listed building had fallen into a state of disrepair and had been described as a target for vandals, fireraisers and drug-users.
The mansion house at Kirkton was built in the middle of the 18th Century, with a number of extensions added over the years.
From the late 1960s until about 2000 it was owned by the Leonard Cheshire foundation.
Plans for a housing development at the site were rejected and earlier this month the developers sought a council review of the decision.
The London Underground's Northern Line extension to Battersea has been given the go-ahead.
[NEXT_CONCEPT]
An attempt to trigger a referendum on Cardiff having an elected mayor is going to fail, a campaigner has admitted.
[NEXT_CONCEPT]
Raith Rovers have signed Martin Scott after the 28-year-old midfielder was released by Scottish Championship rivals Livingston.
[NEXT_CONCEPT]
A derelict historic house near Dumfries has been damaged in a fire which broke out overnight on Friday.