With Manchester United's victory over Liverpool earlier in the day, which put them six points clear at the top of the table, both sides were eager to get more than a point from the match - but were also desperate not to lose.
Chelsea began in more cautious fashion, allowing Arsenal plenty of time on the ball, while laying in wait in their defensive third to snuff out the attacks.
With Robin van Persie and Emmanuel Eboue hugging the touchlines, Arsenal used the width to good effect but without really penetrating.
Dutchman Van Persie sliced one shot high and wide when well placed and drove in a volley that was comfortably saved by Carlo Cudicini.
But while Arsenal were on top with their short passing game in the first 20 minutes, Chelsea's more direct approach dictated the play in the second part of the first half.
A lively Drogba sparked the change and he should have scored when clean though on goal but he miscontrolled the ball.
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As the game opened up, Chelsea scrambled clear after two goalwards toe-pokes from Emmanuel Adebayor and Eboue, and Gallas hit the upright from five yards after sliding in at the back post.
Back came Chelsea and Salomon Kalou seemed set to score but missed his kick in the penalty area before the ball was hacked away and Manuel Almunia just managed to divert Joe Cole's dangerous low cross.
The desire to win the game was obvious from both sides and straight after the interval Matthieu Flamini was prevented from giving Arsenal the advantage when his vicious volley was superbly stopped by Cudicini.
Chelsea continued with their route-one approach and Drogba was through again after a long pass by Claude Makelele but Gallas recovered to do enough to put him off.
And then Arsenal turned defence into attack, literally, as Sagna beat his marker at the near post to head in from a Fabregas corner to score his first goal for the club.
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But the Frenchman's delight soon turned sour when he was forced off with an ankle injury - and suddenly Chelsea stepped up through the gears.
After a couple of wasted chances, most notably a scuffed shot from Michael Ballack, another long pass paid dividends as the ball eventually fell to Drogba, who lashed in a low drive from the edge of the box.
The impetus was now all with Chelsea and when substitute Anelka headed on another long ball, Drogba was there to flick the ball beyond Almunia, who could not get a strong enough hand to it.
It proved enough to secure three points for Chelsea and close the gap on United to five points, while inflicting only a second league defeat of the season on Arsenal.
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