The visitors took the lead when Bacary Sagna headed home Cesc Fabregas' corner on 59 minutes.
But Drogba's late brace put a major dent in Arsenal's title ambitions and resurrected Chelsea's.
Avram Grant's selection of Michael Essien at right-back at the expense of Juliano Belletti was the main surprise for the Blues, while Arsene Wenger chose his strongest line-up.
Chelsea's first attack came via a free- kick on five minutes after Drogba was upended by William Gallas. From the resultant kick, the Ivorian forced a fine diving save from Manuel Almunia.
Arsenal's intricate one-touch football was on display without them troubling Carlo Cudicini in the opening phase of the match.
When the visitors did go close on 16 minutes, Robin van Persie shot wide from just outside the Chelsea penalty area.
Chelsea broke quickly seconds later resulting in Michael Ballack's left-footed shot, on target, again being held by Almunia.
Chelsea were almost the architects of a goal against them when Cudicini headed the ball into the path of Emmanuel Eboue, but his looping half-volley from 35 yards went wide of the empty net.
A snap-shot by van Persie, on 24 minutes, forced Cudicini into a fine, low diving save to his left.
Seconds later a long ball by Ashley Cole bisected the Arsenal defence but Drogba, running free, failed to control the ball and it bounced off his knee in to the gleeful hands of Alumina.
Chelsea were lucky to get away with an offside against Gallas five minutes before the break when the Frenchman stole to the far post but hit the ball onto a post before the linesman's flag went up.
Salomon Kalou hit an air shot as Chelsea moved to the other end with a quick counter-attack.
Ballack again forced a fine save from Almunia on 42 minutes with a rasping left-footer that was held by the Spanish goalkeeper at his left-hand post.
Cudicini produced a fine save from Mathieu Flamini two minutes into the second half, blocking the Frenchman's fierce shot from just inside the Chelsea penalty area.
Gallas made a last-ditch challenge on Drogba on 55 minutes to prevent the striker from getting his shot away.
A wonderful counter-attack by Arsenal two minutes later saw them keep possession of the ball from their own area to Chelsea's. Unfortunately for the visitors, Emmanuel Adebayor's final shot was very weak and was easily held by Cudicini.
The Gunners' breakthrough came a minute later when a Fabregas corner was met by Sagna at the near post as he glanced the ball into the back of the net after getting ahead of his marker Kalou.
Ashley Cole could have levelled within a minute but failed to connect with a deep Joe Cole cross and the ball fell feebly wide.
Grant was forced to make two substitutions on 70 minutes, bringing on Nicolas Anelka for Claude Makelele and Belletti for Ballack and changing formation to a 4-4-2.
Chelsea fans sang: "You don't know what you're doing," to Grant adding "Jose Mourhino" for good measure.
But within seconds Chelsea equalised through Drogba who buried a shot from just inside the penalty area after a long ball by Ashley Cole fell invitingly to the striker.
The turnaround was complete with eight minutes remaining when Anelka headed a ball on from Joe Cole to Drogba who swivelled and volleyed home a low right footer which Almunia could only push into the net.
Drogba almost had his hat-trick but his low drive was parried for a corner by Almunia.
But Chelsea saw out the time, saw off Arsenal and now have Manchester United firmly in their sights.