But once the visitors had gone in front the result seldom looked in doubt and Celtic's challenge rather fizzled out.
After a scrappy opening, Andrey Arshavin popped the ball into the net following a rare flowing move on 11 minutes but the Russian forward had strayed offside.
A promising Celtic break was thwarted by Thomas Vermaelen, with the new Gunners defender showing great pace and awareness to get back and force Scott Brown to skew wide after Aiden McGeady had picked out his team-mate rushing into the penalty area.
Robin van Persie was inches away from gathering a lofted pass over the top from Arsenal skipper Fabregas and the scurrying Arshavin had the sting taken out of his shot by defender Glenn Loovens, allowing Artur Boruc to make a simple save.
Celtic continued to press and a drilled effort from Brown was smothered by Manuel Almunia as it skidded through the legs of lone striker Georgios Samaras, who was blocking the Arsenal keeper's field of vision.
Just as a whirlwind first half looked destined to end goalless, the visitors took the lead thanks to a large slice of fortune.
Aiden McGeady looked lively for Celtic in the first half
Fabregas rifled in a free-kick from 25 yards and the ball deflected off the back of Gallas and spun into the bottom corner of the net, with Boruc wrong-footed.
With Celtic jittery in defence, Arsenal made a strong start to the second half and Denilson curled a shot narrowly wide before the visitors enjoyed another lucky break.
Shaun Maloney's free-kick into the penalty area was diverted on to the arm of Nicklas Bendtner but referee Massimo Busacca waved play on, much to the anguish of the vociferous home support.
Celtic introduced strikers Scott McDonald and Marc-Antoine Fortune for the final half hour and an important sliding block from Gallas prevented the substitutes from combining in front of goal.
With 20 minutes remaining, Arsenal stretched their lead and again it was unfortunate on the home side as Caldwell's outstretched leg diverted a low Clichy cross beyond Boruc.
It may have been an ugly finish but the sweet passing move involving substitute Abou Diaby and the galloping French full-back put the Celtic captain under pressure, with Bendtner waiting to pounce.
Boruc later made an unorthodox block to keep out a venomous strike from Fabregas as Arsenal found more room to exploit and played out the closing stages comfortably.
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