Goals from defender Efe Ambrose and forward Georgios Samaras made it 5-0 on aggregate and confirmed the Hoops face Elfsborg in the next round after the Swedish side won 4-0 against Daugava in Latvia earlier in the day.
Ambrose, who had cut short his delayed summer break for last week’s first leg in Belfast when manager Neil Lennon found himself short of defenders, put Celtic ahead in the 16th minute when he headed in an Anthony Stokes cross after clever footwork from Mikael Lustig on the right.
As the procession towards the Reds goal continued in the first half, James Forrest skimmed Conor Devlin’s bar with a right-footed drive in the 38th minute.
However, there was no addition to the score until the 70th minute when Samaras, who had scored last week in Belfast, made it 2-0 when he headed in a Kris Commons cross from close range.
Gary Hooper — expected to move leave the Parkhead club, probably for QPR — replaced Beram Kayal and hit the outside of the post with an overhead kick but the visitors held out to leave Glasgow with a respectable scoreline.
Elfsborg, well into their league season, will provide a tougher for Celtic test than Cliftonville, whether or not Mo Bangura, on loan from the Hoops, plays against his parent club.
Cliftonville brought more than 1,000 fans for the game. Reds manager Tommy Breslin admitted the match was in reality a damage-limitation exercise for his side but there appeared little they could do to prevent the home side turning the match into a game of attack versus defence in the early stages. In the fourth minute visiting keeper Devlin made a fine save from a Samaras header.
Devlin had no chance with Ambrose’s deadlock-breaking header from Stokes’ cross.
In the 26th minute Devlin took no chances with a Commons thunderbolt from 25 yards, opting to punch clear of the box.
Moments later, following a Cliftonville corner, Hoops defender Kelvin Wilson — a target of Nottingham Forest — stuck his head in the way of a powerful, goal-bound Marc Smyth drive to prevent a shock equaliser.
In the 33rd minute Kris Commons hit another powerful drive from the edge of the box, before Forrest beat Devlin only to see his shot skip off the top of the bar.
Stokes missed with three attempts in quick succession after the break and then, after a fine move by the visitors, Wilson took the ball off the toes of Boyce almost on the goalline to prevent an unlikely equaliser.
Samaras’ goal put a sheen on the scoreline before Amido Balde replaced Forrest to make his competitive debut as Celtic eased home.
CELTIC: Forster, Lustig, Wilson, Ambrose, Izaguirre, Forrest (Balde 76), Kayal (Rogic 65), Brown, Commons, Samaras, Stokes (Hooper 65).
CLIFTONVILLE: Devlin, Cosgrove (O’Carroll 76), Marc Smyth, McGovern, Seydak, Caldwell (Curran 58), Catney, McMullan, Johnston, Garrett, Boyce (Gormley 85).
Ref: Thorsten Kinhofer (Germany).