Steve McClaren's side kept the Gunners at bay for over an hour at Vitesse's Gelredome stadium but Gallas bundled home from close range in the 63rd minute before Adebayor slammed in a second eight minutes from time.
With the second leg at The Emirates to come, this result should be enough to see Arsene Wenger's men through to the Champions League proper, but it was far from a comfortable evening.
Roman Denneboom headed the first chance of the night just wide early on and really should have put Twente in front in the 25th minute when he broke into the area only to fire wildly over.
Marko Arnautovic missed another good chance when a poor touch allowed Manuel Almunia to save from him on the edge of the box and the Dutchmen were almost made to pay before half-time when Gael Clichy's 25-yard effort was paried to the feet of Bacary Sagna - who should have done better from six yards.
Arsenal came more into the game after half-time, though, and went ahead three minutes after the hour mark when Gallas sidefooted home from close range after getting on the end of Robin van Persie's free-kick.
Arnautovic sent another Twente chance into the side-netting in the 79th minute and he was made to rue it three minutes later when Adebayor took Theo Walcott's pass and slammed home for his first of the season.
It will now take a remarkable collapse in London to prevent the Gunners from entering the draw for the Champions League.