Emile Smith Rowe scored the only goal as Arsenal boosted their hopes of European qualification with their first Premier League win at Chelsea for 10 years.
Kai Havertz shot over for the hosts early on, before Smith Rowe capitalised on a Jorginho error on 16 minutes.
The midfielder’s stray backpass was clawed off the goalline by Kepa Arrizabalaga, but Pierre-Emerick Aubameyang then pounced and cut back for Smith Rowe to finish.
Mason Mount had one shot blocked, and another saved by Bernd Leno, as Chelsea rallied.
After the break, Chelsea substitute Olivier Giroud shot wide while Arsenal’s Hector Bellerin forced a save from Kepa.
The Blues nearly snatched a point in stoppage time as Kurt Zouma and Giroud both hit the crossbar in succession, but the Gunners held on.
Arsenal move up to eighth on 55 points, one behind Tottenham Hotspur and two adrift of Liverpool in sixth, although the Gunners have played more matches.
Chelsea stay fourth on 64 points, two behind third-placed Leicester City.