The goal enlivened a staid game that looked certain to be going to penalties as two tired teams slugged it out.
But the first FA Cup Final at the new Wembley Stadium was won by the same team that secured the last Final at the old Wembley, Chelsea having beaten Aston Villa in 2000.
It was the Blues' third victory in the last decade, they also beat Middlesbrough 2-0 in 1997.
However, it was a very humdrum game between the two best sides in England with the first 45 minutes being particularly grim viewing.
On what appeared to be a difficult surface, the first half was played at a very slow pace with chances at a premium as defences dominated.
Both sides struggled to find any space, particularly in midfield, as Jose Mourinho's team struggled without the combined talents of the injured Michael Ballack, Andriy Shevchenko and Ricardo Carvalho.
United stepped up the pace at the start of the second half with Wayne Rooney embarking on two solo runs that warranted greater reward.
His determination seemed to galvanise his team-mates though and, after Petr Cech had parried Rooney's 47th-minute effort, Ryan Giggs lashed a volley just over the bar after good play by Paul Scholes.
When Chelsea hit back, Drogba's deflected free-kick on the hour hit the outside of the post before Cech raced off his line to dive at the feet of the rampaging Rooney.
United appeared to have the edge in extra-time and almost grabbed a 103rd-minute winner.
Rooney's tantalising cross seemed certain to be knocked home by Giggs, but the veteran Welshman failed to get a good connection on the ball six yards out and Cech was able to grasp the ball.
As Giggs' momentum took him into the keeper, the ball appeared to cross the line, but no goal was awarded.
Penalties seemed certain until, with four minutes of extra-time remaining, a rare piece of quality finally broke the deadlock.
Drogba and Frank Lampard exchanged intricate passes which allowed the Ivorian some space inside the penalty area. He used it perfectly clipping the ball over Edwin van der Sar and into the net.
The goal deflated United so Mourinho's side end the season with a Carling and FA Cup double while Sir Alex Ferguson will have to settle for another Premiership title.