The Gunners, beaten by Manchester City and Chelsea in their first two outings, bounced back through a Nacho Monreal strike, an Issa Diop own goal and a Danny Welbeck tap-in after Marko Arnautovic had given United a surprise lead.
A failure to hold on means that the Hammers, who will feel that they were worthy of a point, have now lost their opening three league fixtures for the third time this decade.
Manuel Pellegrini handed full debuts to Diop and Carlos Sanchez, but it was another of his summer signings who proved key early on as Lukasz Fabianski got right behind Henrikh Mkhitaryan's well-struck shot.
A better chance went begging up the other end soon after as Arnautovic fired into the side-netting when played through - the first of countless occasions that the Hammers failed to take full advantage of their opponents' high defensive line.
That setback managed to awake Arsenal as, a week on from dragging themselves back into the game from two goals down at Chelsea, they bounced back four minutes later here through a simple finish from Monreal towards the back post.
Hector Bellerin played a big part in the goal with his cross from the right after getting in behind, and that was proving to be a big theme as a couple more big chances came and went up the other end.
Michail Antonio was played through but elected to shoot early, only finding Cech with his attempt, while Robert Snodgrass also met his match in Cech from six yards out with only the goalkeeper to beat.
The introduction of Alexandre Lacazette for the second half boosted the hosts in an attacking sense, with the Frenchman getting a couple of early shots on target and Shkodran Mustafi directing a header on goal that Fabianski did well to claw away.
United always looked threatening, however, and they could easily have been in front on the hour when Anderson powered down the left and picked out Arnautovic, who fired the ball at Cech down the middle.
The vital third goal of the match went the way of Arsenal after Lacazette's cross hit Diop in the midriff and crossed the line for a debut own goal.
An Arsenal onslaught followed, with Aubameyang and Lacazette both coming close, before Aaron Ramsey's drive was kept out by another fine Fabianski save on his return to this ground.
Those missed opportunities almost came back to haunt the hosts as their former striker Lucas Perez failed to convert from a good position late on, neither finding the target with his shot nor picking out one of his teammates at the end of a swift counter.
Substitute Welbeck then wrapped up the three points for the Gunners in added time by keeping his cool in front of goal after being spotted by Monreal in acres of space.
ARSENAL (4-2-3-1): Cech; Bellerin, Mustafi, Sokratis, Monreal; Guendouzi (Torreira 56'), Xhaka; Iwobi (Lacazette 46'), Ramsey, Mkhitaryan; Aubameyang (Welbeck 75')
WEST HAM UNITED (4-2-3-1): Fabianski; Fredericks, Balbuena, Diop, Masuaku; Sanchez, Wilshere; Anderson, Snodgrass (Yarmolenko 75'), Antonio (Perez 63'); Arnautovic (Hernandez 59')