Drogba produced a rare moment of brilliance in a game that failed to live up to expectations to complete a cup double for the Blues.
He played a one-two with Frank Lampard before stabbing past Edwin van der Sar with the tie headed for penalties.
Ryan Giggs missed a good chance to win the game for United in normal time.
The 90,000 fans inside the stadium were treated to a fantastic pre-match build-up featuring some famous faces from Wembley's glorious past.
But aside from Drogba's late winner, the two best sides in the country produced a game that is unlikely to live long in the memory.
That is unlikely to bother Chelsea, who won the last FA Cup final at the old Wembley in 2000.
A cagey opening 20 minutes set the pace for a match that at times seemed more like the Community Shield than the first FA Cup final at an immaculate new Wembley.
The first effort on target did not arrive until after the half hour with Paulo Ferreira venturing forward and slipping a pass in for Frank Lampard, who forced Edwin van der Sar to make a smart low save.
That was about the extent of the first 45 minutes as both teams looked heavy-legged after a season in which they have each played over 60 games.
The pitch, which looked in fabulous condition at the start, began to cut up which did not aid the footballing spectacle or the weary players.
Joe Cole looked one of the few capable of livening things up before he was surprisingly replaced at half-time by Arjen Robben.
The pace picked up slightly after the break, with United looking the more likely to get Wembley's first Cup final goal.
Wayne Rooney emerged from his first-half shell to bring a first save out of Petr Cech and tested Chelsea's backline with a couple of bursting runs.
Paul Scholes picked out Giggs to volley just over the top as Sir Alex Ferguson's men showed the odd glimpse of the stylish side that dominated the Premiership.
At the other end Drogba was largely contained by Rio Ferdinand and Nemanja Vidic, with the Premiership's top scorer limited to a curling free-kick that clipped the outside of the post during the 90 minutes.
Vidic could have settled the tie with the last touch of normal time but his glancing header from Giggs' inswinging free-kick was just too high.
Both sides had glorious chances to break the deadlock at the end of the first period of extra-time.
Giggs somehow managed to pick out Cech with the goal at his mercy as he slid onto a Rooney cross and then Drogba headed into the side-netting from an equally-promising position.
Salomon Kalou was agonisingly close to scoring before Drogba and Lampard exchanged passes and the Golden Boot winner capped a fabulous season with the most important goal of his season.
It prevented a third successive FA Cup final going to penalties and provided a memorable moment for the long-awaited return home for the FA Cup.
Chelsea add the FA Cup to the League Cup which they won earlier this season and manager Jose Mourinho did his job prospects no harm by securing the one domestic trophy that was missing in his collection.
Edit: More pics from the BBC
Prince William officially opened the new Wembley
Both teams walk onto the pitch
Chelsea celebrate their win, with John Terry (left) and Frank Lampard pictured
A disappointed Manchester United team (with a metrosexual-as-ever Ronaldo)
Chelsea manager Jose Mourinho lifts the cup
Thanks to plc4miehaed for the other pics