The Gunners were down to 10 men after Gabriel Paulista was sent off at the end of the first half.
Chelsea took advantage of the extra man and took the lead early on in the first half. The Blues secured victory late on when Eden Hazard’s shot was deflected into the back of the net.
Jose Mourinho’s side moved up to 12th place while Arsenal remain in forth position with ten points in six games.
The Gunners take on their rivals Chelsea in the League Cup on Wednesday.
Chelsea 2 -0 Arsenal
Stadium: Stamford Bridge
Begovic; Ivanovic, Zouma, Cahill, Azpilicueta; Matic, Fabregas; Pedro, Oscar, Hazard; Diego Costa.
Subs: Blackman, Terry, Mikel, Ramires, Loftus-Cheek, Falcao and Remy
Cech; Bellerin, Gabriel, Koscielny, Monreal; Coquelin, Cazorla, Ramsey, Ozil, Sanchez; Walcott
Subs: Ospina, Debuchy, Gibbs, Chambers, Arteta, Chamberlain, Giroud