The Scottish league winners were beaten 6-1 on aggregate by Legia Warsaw in the previous round but the Poles were punished for fielding an ineligible player.
Bartosz Bereszynski, who came on as an 86th-minute substitute as Legia won the second leg 2-0 at Murrayfield on Wednesday night, was suspended following a red card for violent conduct in last season's Europa League.
Hours before the draw for the next round was to be made, UEFA announced that Celtic had been awarded a 3-0 win in the second leg, and therefore qualified on away goals after a 4-4 aggregate result.
Celtic reached the group stages last season but failed to progress from an extremely tough pool that also included AC Milan, Barcelona and Ajax.
Arsenal will meet Turkish side Besiktas in the two-legged play-off.
Arsene Wenger's side, who finished fourth in the Premier League last season, will travel to Istanbul for the first match in the week beginning August 18, with the home leg taking place the following week.
Arsenal have never previously met Besiktas in any European competition.