The Ivory Coast striker pounced on Frank Lampard's lofted pass with six minutes remaining to give Chelsea their first FA Cup success over Arsenal since 1947.
They will face the winners of Sunday's semi-final between Manchester United and Everton, which is also at Wembley.
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PHOTOS: Arsenal 1-2 Chelsea
Drogba has now scored eight goals against Arsenal since joining Chelsea from Olympique Marseille, including a late winner in the League Cup final in 2007.
Neither side were at their best following midweek triumphs in the Champions League and Chelsea had to dig deep after Theo Walcott volleyed Arsenal ahead in the 18th minute.
Chelsea's equaliser came from the unlikely source of winger Florent Malouda before the break.
Arsenal had the better second-half chances before some hesitation by second-choice goalkeeper Lukasz Fabianski let in Drogba who slotted home from a tight angle.
Interim manager Guus Hiddink can now sign off his short reign at Chelsea with an FA Cup final showdown on May 30.
"This is a huge step to be in the FA Cup final because it's so respected worldwide," said Hiddink.
The Dutchman's only other appearance at the new Wembley ended in defeat when his Russia side lost 3-0 to England in a Euro 2008 qualifier. "I think we deserved it today. We adapted well after they scored and we started to control and dominate. The sloppy start was down to a tactical reason," said Hiddink.
"I'm very proud of these players, they always deliver, they always react to a setback."
Hiddink's Chelsea side appeared to be suffering from the physical effects of their epic 4-4 draw with Liverpool on Tuesday as Arsenal began brightly on Wembley's lush turf.
Apart from one moment early on when Drogba's header was cleared off the line by Arsenal right back Kieran Gibbs, it was the north London side who looked brighter.
QUALITY MOVE
The first move of real quality put Arsenal in front. Gibbs, deputising at left back for the injured Gael Clichy, combined well with Emmanuel Adebayor and his cut back was met on the volley by Arsenal's other young Englishman Walcott.
His strike was not convincing but a brush off the hand of Chelsea defender Ashley Cole meant it bounced awkwardly and evaded the diving Petr Cech.
Chelsea were level after 33 minutes with the hard-working Lampard, their best player, delivering an inch-perfect diagonal pass to the foot of Malouda.
The big-money signing from Olympique Lyon beat Emmanuel Eboue with e
