Hammersmith and Fulham council's planning committee have backed plans to demolish the current 41,600-capacity Stamford Bridge stadium.
The plans include a walkway from the nearby District Line station.
"We are grateful that planning permission was granted for the redevelopment of our historic home," Chelsea said in a statement.
"The committee decision does not mean that work can begin on site. This is just the latest step, although a significant one, that we have to take before we can commence work, including obtaining various other permissions."
London Mayor Sadiq Khan will have the final say on whether Chelsea can build their new stadium.
The new stadium has been designed by architects Herzog and de Meuron, who were also responsible for the "Birds Nest" Olympic stadium in Beijing.
The proposals could mean owner Roman Abramovich has to find a temporary home for the current Premier League leaders for up to three years, with both Twickenham Stadium and Wembley Stadium being looked at as possible options.
Chelsea might, however, struggle to use Wembley as north London rivals Tottenham Hotspur will occupy the national stadium for at least the 2017-18 football season as work finishes on Spurs' own new 61,000-capacity stadium.
Chelsea could stay at Stamford Bridge while the work takes place but this is thought to be the most expensive option.
Mr Abramovich has wanted to increase capacity at Chelsea on match days for a number of years.
He previously attempted to buy Battersea Power Station with a view to redeveloping the site into a new stadium, ultimately losing out to property developers who are currently building luxury apartments at the site.
Ten years ago Arsenal built the 60,000-seat Emirates Stadium, last summer West Ham moved to the 57,000-capacity Olympic Stadium in Stratford, east London, and Spurs are currently redeveloping their White Hart Lane ground.
The current 41,663-capacity Stamford Bridge is the seventh biggest stadium used by a Premier League team, well behind Manchester United's 76,000-seater stadium at Old Trafford.

Jack Cottle, of Durgates, Wadhurst, East Sussex was arrested after footage of a VW Polo being driven into a Fun Cup race on 14 June was posted on YouTube.
He admitted causing a public nuisance by driving on to a live racing circuit causing it to be stopped and putting competitors in harm's way.
Cottle is to be sentenced in November.
Judge Martin Joy told him: "Motor racing is an inherently dangerous sport and quite plainly it was a matter that put persons at risk for their lives.
"You must understand that the court will be considering a custodial sentence."
The white car, carrying two passengers in addition to Cottle, was seen in the video travelling along the Kent track at high speed as other cars passed by during the last 30 minutes of the four-hour race.
The footage has been viewed more than a million times online.
Prosecutor Keith Yardy told the court: "As a result of his actions, the race was brought to a stop. He was interviewed. He said he was dared to do it and was sorry but would do it again."
However, defence counsel Ailsa Williamson said Cottle was "an immature man" but had the support of his family who attended court with him.
She said: "He was dared by friends. This was a foolish prank that should never have gone this far and he deeply regrets his actions."

Episode VII: The Force Awakens will now be released in Britain on 17 December, a day earlier than previously announced.
The latest film is the first in a new trilogy for the franchise.
It will will run alongside the original three films and prequels that followed them.
The Force Awakens will be set 30 years after Return of the Jedi and the trailer sees the return of Han Solo, played by Harrison Ford again, alongside co-pilot Chewbacca.
French film fans get to see it before anyone else with the release date in France set for 16 December.
Chelsea Football Club have been given permission by the local council to build a new £500m 60,000-seat stadium.
[NEXT_CONCEPT]
A 22-year-old man who drove his girlfriend's car on to the Brands Hatch circuit during a race could face jail.
[NEXT_CONCEPT]
The UK release date for the new Star Wars film has been brought forward by a day meaning it will be out here a day before it hits cinemas in America.