Adebayor forced home defender Julien Faubert to concede an own goal, before clinching the points with a clinical second goal in a frantic finale that also saw Hammers' striker Carlton Cole dismissed.
Arsenal made a trio of changes following their 5-2 Champions League victory over Fenerbahce, as William Gallas, Robin Van Persie and Nicklas Bendtner each came in for Denilson and substitutes Abou Diaby and Adebayor.
And from the very first minute, Arsene Wenger's men looked set to carry on from where they left off in Turkey in midweek as Theo Walcott soon steered a low, angled 12-yarder into the arms of the well-placed Robert Green before Bendtner drilled wide from 18 yards.
Walcott's wizardry down the right-flank was causing the Hammers all sorts of problems and, on 20 minutes, the electrifying England international burst clear from halfway and sent a rising 15-yarder crashing skywards off Green's left-hand angle.
Moments later, the Hammers' keeper raced from his line to intercept from the breaking Bendtner, whose vociferous claims that the ball had been claimed inches outside the area fell on the deaf ears of referee Phil Dowd.
Television replays suggested that Green had been fortunate to escape a red card, but there was nothing lucky about his two subsequent, superlative saves that saw him palm Walcott's point-blank shot aside, before denying Bendtner with an equally agile stop.
Gianfranco Zola had responded to last Sunday's defeat at Hull City by making four changes to his eighth-placed team as David Di Michele, Hayden Mullins, Lee Bowyer and James Collins replaced Matthew Etherington, Valon Behrami, Mark Noble and skipper Lucas Neill.
And while West Ham spent most of the first half firmly on the back foot trying to contain Cesc Fabregas, Van Persie and Walcott, their tenacious tackling and opportunist offensive play still gave the Gunners some uncomfortable moments, too.
Indeed, Di Michele saw his dipping 25-yarder brilliantly diverted into the East End skies by Manuel Almunia and when breaking Craig Bellamy showed both Mikael Silvestre and Emmanuel Eboue a clean pair of heels, the Arsenal keeper diverted the Welshman's low shot over the bar with his outstretched right leg.
As the hour mark came and went, the deadlock still had not been broken and midway through the second period Wenger introduced both Diaby and Adebayor at the expense of Samir Nasri and, surprisingly, Walcott.
Within seconds of his arrival Adebayor headed inches wide and then Green produced another brilliant stop after Van Persie sent a low 20-yarder scorching through a pack of bodies.
On 70 minutes, Van Persie was left looking to the heavens, yet again, after his 25-yard free-kick beat the flying Green only to crash back off the relieved keeper's right-hand post.
But just when it looked like it was not going to be the Gunners' day, West Ham inadvertently handed them the lead with just a quarter-hour remaining.
Adebayor danced his way through the Hammers' area before steering a weak shot across the six-yard box, where the extended leg of the flustered Faubert prodded the ball beyond Green.
And then in the final minute, with West Ham pressing forward looking for an elusive equaliser, Adebayor raced onto Bendtner's diagonal throughball and rounded the stranded Green before firing an 18-yarder into the unguarded net.
It was a miserable end to an afternoon that had looked like yielding a hard-fought draw for Zola's men, whose day got worse in stoppage-time when Cole was harshly sent off for sliding through Alex Song.