The Gunners should have opened the scoring inside five minutes, but Pierre-Emerick Aubameyang squandered a good chance from six-yards out after some good work by Alexandre Lacazette
However, Arsenal went in front in the 13th minute through Lacazette's superb finish in the box, although the Chelsea shot-stopper was beaten from a tight angle.
Unai Emery's men were given a new lease of life in the first half and they certainly took advantage when Laurent Koscielny doubled the lead in the 39th minute.
The woodwork came to Arsenal's rescue just before the halftime break when Blues defender Marcos Alonso's header came off the post.
The second half continued in similar fashion to the first, but there weren't as many chances in the respective final thirds.
The Gunners were dealt a serious blow in the 65th minute when Hector Bellerin was stretchered off after a serious knee injury.
As the second half went on, chances were hard to come by on both ends despite the Blues seeing more of the ball.
That was the trend for the remainder of the second half as the Gunners reclaim fifth place and are now six points behind the Blues, who miss the chance to move third.