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.
NEWS: Hiddink hopes for perfect Chelsea finale
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.
Chelsea's equaliser came from the unlikely source of winger Florent Malouda before the break.
"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.
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.
