Drogba caressed Ashley Cole’s low cross into the net at the near post five minutes before the break for his 13th goal against Arsene Wenger’s side in 11 starts.
Defender Alex settled the contest with a powerful 30-yard free-kick in the 85th minute to hand the Gunners their second successive league defeat.
The tempo was set when Arsenal almost took the lead in the opening minute, with Bacary Sagna’s right-wing cross headed just wide by Marouane Chamakh.
Referee Mike Dean adjudged that the ball had deflected off a Chelsea defender and when the corner found its way into the six-yard box, Lauren Koscielny somehow headed over from two yards out.
It was a wake-up call for the home side who responded with a Drogba free-kick that ballooned over the crossbar.
In the ninth minute, Chelsea almost opened the scoring when John Terry found Florent Malouda in the inside left channel, but the France international’s audacious chip was just over.
Arsenal hit back immediately with Andrey Arshavin forcing Petr Cech into a save when he cut inside from the left wing and unleashed an angled drive.
In the 19th minute Chelsea squandered another chance to open the scoring when Michael Essien rose unmarked in the six-yard box but headed Drogba’s corner wide of the target.
Cech saved Chelsea in the 28th minute when he dived to his left to tip Arshavin’s fierce 20-yard drive around the post.
Arsenal were enjoying as much of the possession as their hosts and Samir Nasri was a yard wide after outwitting the Chelsea defence on the edge of the penalty area.
Chelsea’s frustration was growing and it was made worse then Drogba and Malouda were spotted arguing with each other in midfield.
But it clearly had an effect as in the 34th minute Lukasz Fabianski pulled off a great save to deny Drogba after the Ivory Coast striker had been put in the clear by Essien.
It was end-to-end stuff and a typical fast-flowing London derby that had everything but a goal – until the 40th minute. It was no surprise when the scorer turned out to be Drogba.
It was a superb move which led up to his finish. A quick free-kick eventually saw Ramires put Cole through on the left and when the former Arsenal full-back crossed the ball, Drogba sidefooted it into the net at the near post.
The Gunners should have equalised in the 58th minute when Chamakh opted to take one touch too many on the edge of the penalty area and Ramires was able to block the danger.
But Chelsea should have made it 2-0 moments later when Anelka robbed Sebastien Squillaci and rounded Fabianski, but the France international sent his shot into the side-netting.
In the 64th minute, Cech was alert enough to take the ball off the feet of Chamakh after Nasri’s clever ball looked to have put the Arsenal man in with a clear opportunity to score.
Arsenal replaced Abou Diaby with Tomas Rosicky in the 71st minute as they looked for more invention in front of goal – and Rosicky soon forced Cech into a fine save.
However, Chelsea wasted another chance to kill off the game in the 73rd minute when Anelka raced clear of the Arsenal defence but his shot was somehow kept out by Fabianski.
Chamakh failed to hit the target from six yards when he sent a firm header just past the right-hand upright with 10 minutes left – and five minutes later Alex put the outcome beyond doubt when he crashed a 30-yard free-kick into the roof of the net.