Cheikhou Kouyate put the Hammers ahead late in the first half when he beat Cech to a free-kick and headed into the empty net.
Cech, a summer signing from Chelsea, was beaten again in the 57th minute when he was wrong-footed and unable to keep out Mauro Zarate’s shot from outside the box.
The Gunners hardly threatened a comeback as the visitors, for whom 16-year-old Reece Oxford started in midfield, held out to secure all three points in Slaven Bilic’s first top-flight match in charge.
The Croatian could hardly have wished for a better performance on the opening day after seeing his side knocked out of the Europa League in Romania on Thursday evening.
Bilic sprang a surprise in his starting line-up at the Emirates by handing Oxford his league debut, making him the seventh-youngest player to feature in the Premier League.
Olivier Giroud was given the nod in attack for Arsenal ahead of Theo Walcott and Alexis Sanchez made the bench even though he was not expected to feature.
Arsenal, who won the FA Community Shield at Wembley last weekend with victory over Chelsea, made a bright start with the lively Alex Oxlade-Chamberlain volleying the ball over the crossbar from 20 yards.
West Ham offered some attacking threat of their own, with Zarate and Diafra Sakho at the top of the diamond fed by the impressive Frenchman Dimitri Payet.
The closest either side came to an opener was in the 31st minute when Aaron Ramsey’s 20-yard half-volley deflected off the back of defender Aaron Creswell and looped onto the crossbar.
But the visitors took the lead late in the half when Payet whipped a free-kick into the right side of the Arsenal penalty area and as Cech rushed off his line in an attempt to punch clear, he was beaten to the ball by Kouyate, who headed into the unguarded net.
The Gunners laboured through the opening stages of the second half before Giroud fired into the side netting after being played down the left of the penalty area by Mesut Ozil.
But it was West Ham who struck next after 57 minutes as Oxlade-Chamberlain failed to clear his lines properly and Zarate picked up the loose ball, turned and fired a shot into the bottom corner past a wrong-footed Cech.
Wenger responded by bringing on Walcott and then Sanchez but the Gunners continued to struggle to create chances.
Oxlade-Chamberlain smashed the ball into the side netting and Sanchez forced a save from Adrian in injury-time.
The hosts were unable to force their way back into the match, though, as West Ham closed out a first victory over the Gunners since April 2007.
Arsenal: Cech (5), Debuchy (5), Mertesacker (6), Koscielny (5), Monreal (5), Coquelin (5), Ramsey (6), Oxlade-Chamberlain (6), Ozil (5), Cazorla (6), Giroud (6).
Subs: Walcott (5), Sanchez (6).
West Ham: Adrian (7), Tomkins (6), Reid (7), Ogbonna (6), Cresswell (6), Oxford (7), Noble (6), Kouyate (7), Payet (7), Zarate (7), Sakho (7).
Subs: Nolan (5), Jarvis (5), Maiga (5).
Man of the Match: Winston Reid (West Ham)