Arsenal nearly made an ideal start when West Ham defender James Collins was forced to head against his own bar from a corner, but it turned out to be a false dawn for what proved to be a dismal first 45 minutes.
Neither side mustered a serious effort on target, with Arsenal comfortably on top in terms of possession, but doing very little to stretch the West Ham defence or threaten Robert Green's goal.
Wenger praises West Ham's defence
Emmanuel Eboue was industrious and involved in most of Arsenal's attacks, but his afternoon came to a premature end when he limped off, to be replaced by Carlos Vela, following a challenge by Scott Parker.
Collins had West Ham's only vaguely threatening chance with a header in first-half injury-time, and the Welshman was at the centre of the action once again shortly after the restart when he cleared the best chance of the game so far.
Green's poor punch clear only went as far as Kolo Toure, near the edge of the box, and he hooked the ball back over his shoulder towards the goal, but Collins was on hand to head clear.
Twice in quick succession Matthew Upson had to make blocks from Denilson efforts as Arsenal continued to play at a higher tempo, and Adebayor wasted the best chance of the game so far when he poked Bacary Sagna's cross wide of the post.
Hammers hard work paying-off - Clarke
Arsenal continued to pile on the pressure, but another great chance went begging when Adebayor put his free header straight into the arms of Green.
West Ham barely left their own half as they soaked up Arsenal's punches, but Carlton Cole's easily held shot from distance showed the home side they could not afford to be too complacent.
The longer the game went on the more desperate Arsenal became, and the full price of Adebayor's missed chances became clear.
Adou Diaby was stretchered off after a collision with Savio in injury time, which summed up a frustrating afternoon for the Gunners.
Arsenal manager Arsene Wenger:
"We had 70-30 possession but they defended very well as a team.
"We were very close to scoring and gave everything, but couldn't get the goal which would get us the three points.
"You have to continue fighting, usually we get a goal in last 15 minutes when they defend like that us against us."
West Ham coach Steve Clarke:
"It's always difficult when you come here against a team who pass so well, you have to be disciplined and do the right things.
"Maybe I'm disappointed we didn't do more with the ball, but today was more a defensive performance.
"We came here with the intention of playing as we have been playing recently and didn't concede too many clear-cut chances which was pleasing.
"If you want success at a club, you have to start with basics. They're working hard for the manager and themselves, and you can see the result of that on the pitch."
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