With Cesc Fabregas missing, Wenger handed a competitive debut to teenager Aaron Ramsey in midfield alongside Denilson and the youngsters struggled to make an impact in the game.
Worryingly the backline looked suspect as ex-England boss Steve McClaren had his FC Twente side fired up.
Eljero Elia and Denneboom caused plenty of problems for a reshaped defence in the absence of Kolo Toure.
Denneboom headed over the bar in the opening minutes and then went even closer with a rising shot just off target after busting into the box unchallenged.
The Dutchman slipped a delightfully-timed pass through to Arnautovic to send him clean through, but a poor first touch allowed Manuel Almunia to come out and block his effort.
Ramsey showed the confidence of youth with a shot from way outside the box that failed to trouble Sander Boschker.
FC Twente boss McClaren embraces Wenger before the game
The Dutch side's keeper was beaten to the ball in a rare sight on goal for Van Persie, but the Arsenal forward could only knock a square ball into a danger area devoid of team-mates.
There was more urgency from Arsenal after the break but Gallas's goal still came as a surprise, not least to the defender as Van Persie's inswinger landed at his feet a few yards out.
Gallas managed to deflect the ball on target and Boschker could not react quickly enough to keep it out.
Arnautovic gave Arsenal another fright but the young striker, who has yet to break his duck for the first-team, poked into the side-netting.
With both sides tiring, Walcott's pace allowed him to get on the end of a hopeful through ball and he squared for an unmarked Adebayor, whose effort just squeezed under the body of Boschker.
It was anything but a textbook Arsenal victory, but for once Wenger will be happy to have won ugly.
FC Twente manager Steve McClaren:
"I think we should have scored in the first half. You don't get many chances and you have to take them. We didn't do that. We lost concentration on two occasions and got punished.
"That's what happens in Europe against the top teams. The inexperience in the dressing room showed and we have to learn from it.
"But I can't fault the performance. We had Arsenal rattled, especially in the first half."
Arsenal manager Arsene Wenger:
"We expected a very physical game and we were not disappointed.
"We needed to be patient and not make mistakes at the back.
"They dropped their level but to do what they did in the first half is very difficult and from then on we looked more comfortable.
"We were so young in the middle of the park for a game of that level and you could be worried but I have confidence in them."
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