The champions took the lead on 14 minutes thanks to Sterling's low shot from the edge of the area for his 50th Premier League goal.
Petr Cech kept Arsenal in it in the first half with a brilliant double save from Riyad Mahrez’s free-kick and Aymeric Laporte’s follow-up.
Cech also denied Sergio Aguero shortly before Silva swept home City’s second after a swift counter-attack.
Arsenal substitute Alexandre Lacazette fired their best chance narrowly wide of the post.
Arsenal: 18 Aug v Chelsea (A)
Man City: 19 Aug v Huddersfield Town (H)