Gunners boss Unai Emery made seven changes to the side that blew a two-goal half-time lead at Watford at the weekend as David Luiz, Sead Kolasinac, Granit Xhaka and Pierre-Emerick Aubameyang were the only players to retain their place.
Frankfurt, who went into the game unbeaten in their last 16 home matches in European competition, had the first effort of note in the fourth minute when Filip Kostic played in Djibril Sow and his fierce effort was straight at Emiliano Martinez.
Arsenal responded within a minute but Lucas Torreira spurned a great chance when he fired over with the goal at his mercy following good work down the left by Bukayo Saka.
Andre Silva fizzed a shot just wide from outside of the area and Kostic got in behind Calum Chambers but fired into the side-netting as Frankfurt engineered a couple of chances before the 20-minute mark.
Martinez saved Chambers’ blushes on 23 minutes when Kostic again got the better of him, but his subsequent effort was diverted wide off the fingertips of the Gunners’ custodian.
Arsenal gradually worked their way into the contest and had a couple of close attempts as Joe Willock flashed a shot just wide before Emile Smith Rowe was denied in a one-on-one situation against Frankfurt keeper Kevin Trapp on 32 minutes.
Bas Dost sent a free header wide for Die Adler in the 37th minute and he was made to pay when Arsenal surged to the opposite end and went in front.
Saka released Willock down the left and he cut between two defenders before seeing his goalbound effort deflect over Trapp off Abraham and into the net via the underside of the crossbar.
Frankfurt tried to get back on level terms in the latter stages of the half, but again Martinez was on hand to pull off a smart save to deny Andre Silva on 45 minutes.
The hosts came out firing after the interval as Dost sent a header from four yards just wide as he was at full stretch to meet a left-wing cross from Martin Hinteregger, before Andre Silva was just off target on 49 minutes.
Hinteregger was the hero for Frankfurt moments later as he denied Aubameyang with a goal-saving block after the Gunners’ forward was played in by Willock following a swashbuckling 60-yard run.
Arsenal were inches away from extending their lead on 56 minutes but Xhaka’s free-kick into the area bounced off the crossbar after Trapp appeared to lose the flight of the ball.
The game moved from end-to-end as Willock was denied when Trapp came off his line to save at his feet, before Kostic lashed a low strike which flashed across the face of goal seconds later.
The Gunners’ task of holding on to their lead was made easier when Dominic Kohr was sent off for two bookable offences 11 minutes from time following a cynical foul on Saka.
Indeed, the three points were secured on 85 minutes when Saka scored his first goal for Arsenal from 20 yards following a quick counter-attack.
Two minutes later Aubameyang put the icing on the cake as Saka turned provider for the Gabon international who picked his spot inside Trapp’s near post.