Theo Walcott had opened the scoring for the Gunners at Wembley, but Florent Malouda had the Blues level before the interval.
Arsene Wenger's side survived an early scare as a defensive mix-up almost saw Drogba nod into an empty net, but Kieran Gibbs recovered in the nick of time to hack off the line.
The Gunners then began to take control of proceedings and took the lead on 18 minutes as Walcott volleyed home a Gibbs cross via the aid of a slight deflection off Ashley Cole.
Having fallen behind, Chelsea then suddenly sprung into life and Malouda grazed a post with a drilled effort which squeezed under the body of Lukasz Fabianski.
The Frenchman was not to be denied, though, and he found the target just past the half-hour mark when he beat the Arsenal offside trap and fired low into the bottom corner.
It could have been even better for the Blues at the break had Nicolas Anelka taken advantage of a clear sight of goal, but he saw a low drive cannon back off the post.
The second half was a quieter affair in terms of clear-cut goalscoring opportunities, but both sides continued to pile forward whenever the situation allowed.
Emmanuel Adebayor blazed a good chance over for the Gunners inside three minutes of the re-start, while Chelsea proved to be wasteful with their final ball despite working a number of promising openings.
Frank Lampard did his utmost to force a further breakthrough with a controlled first-time effort midway through the second period, but his radar was slightly off and he could only brush the side-netting.
With only six minutes left on the clock it appeared as though extra-time was a formality, with nothing to choose between two Premier League heavyweights.
However, Drogba had other ideas and he coolly rounded the onrushing Fabianski to slot in a dramatic winner and send the Chelsea support into delirium.