Neil Lennon's side recorded a 5-0 aggregate victory over the Irish League champions to reach the third qualifying round of the competition
After a 3-0 win at Solitude, the Bhoys were in full control of the tie and goals from Efe Ambrose and Georgios Samaras help them progress.
Irish striker Anthony Stokes impressed from the start, winning an early corner which saw Conor Devlin producing the first save of the match to deny Georgios Samaras. Devlin was kept busy by the Bhoys, and had to pick the ball out of the net on 16 minutes when Efe Ambrose headed in Stokes' cross.
Celtic continued to dominate, but only went in a goal ahead at the break thanks to good work from Devlin in the away goal, while James Forrest had a couple of good chances go over the crossbar.
Stokes went close to getting his name on the scoresheet early in the second half, but was denied when Devlin got down well to save.
Northern Ireland international Liam Boyce almost scored an unlikely equaliser, but saw his effort cleared off the line by Kelvin Wilson.
The hosts doubled their lead 20 minutes from time with Samaras heading in his second goal in two games, meeting Kris Commons' delivery.
Substitute Gary Hooper hit the post late on as Celtic eased into the third qualifying round, where they will meet Swedish side IF Elfsborg, with the first leg on July 30-31 at Celtic Park.