Goals from defender Efe Ambrose and forward Georgios Samaras made it 2-0 on the night and 5-0 on aggregate and confirmed the Hoops will play Elfsborg in the next phase after the Swedish side won 4-0 against Daugava in Latvia earlier in the day to progress 11-1 over the two legs.
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 - who has been linked with a move away from Parkhead - 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 test for Celtic than Cliftonville whether or not Mo Bangura, on loan from the Hoops, plays against his parent club and that is the issue which will surround the tie until kick-off time at Parkhead next week.
Despite the difficulty if not impossibility of their task, Cliftonville brought over 1,000 enthusiastic and vociferous fans for the game.
Reds manager Tommy Breslin admitted the match was in reality a damage-limitation exercise for his part-time 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.
But it was not all one-way traffic and 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 the Irish supporters having something concrete to cheer.
In the 33rd minute 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 increasingly-confident visitors, Wilson took the ball off the toes off Boyce almost on the goalline to prevent an unlikely equaliser.
Samaras' goal, which put a better sheen on the scoreline, came before Amido Balde replaced Forrest to make his competitive debut and with 10 minutes remaining he headed an Emilio Izaguirre cross past the post.