The tie was Celtic's first-ever competitive meeting with a Northern Irish club and despite the hosts' early resilience, the SPL champions created a number of first-half chances, with goals from Mikael Lustig and Georgios Samaras putting the Hoops in firm control of the game.
Celtic continued to dominate throughout the second half but it was not until late on when Scotland international James Forrest made it three to seal the win and all-but end Cliftonville's hopes of an upset.
Neil Lennon's side hit the woodwork in the opening minutes when Kris Commons' looping header found the crossbar after good work from Forrest down the left-hand side. Forrest then came close to opening the scoring moments later when he cut inside of George McMullan to fire over the bar from inside of the area.
The Hoops created the better openings of the early stages with Anthony Stokes and Samaras both scuppering half-chances, while Beram Kayal came close to breaking the deadlock with a good strike from distance which just evaded the post.
Celtic eventually took the lead in the 24th minute through Lustig when the Swedish defender met Commons' corner with a near-post header to steer past a helpless Conor Devlin in the Cliftonville net.
Cliftonville almost responded within minutes of the opening goal with their first attempt of the night on target, but Liam Boyce's shot on the turn could only find the arms of Fraser Forster.
The visitors then doubled their lead on the half-hour mark thanks to a left-footed strike into the roof of the net from Samaras. Having switched to the right wing, the impressive Forrest picked out Commons inside the area and he teed up the Greek striker who executed a powerful finish.
For the second time on the night, Commons found himself denied by the woodwork just before the break. After jinking past three defenders, the former Nottingham Forest man unleashed a seemingly goal-bound curling effort, but the inside of the post stood in his way once again.
The hosts almost gave Celtic cause for concern shortly after the restart when confusion between Lustig and Forster presented Stephen Garrett with an opening, but he was unable to take advantage of the mix-up as the England goalkeeper managed to recover and preserve his clean sheet.
Celtic dominated possession throughout the early stages of the second-half and Commons should have made it three just after the hour mark when he drove into the box, but his effort was kept out by the legs of Devlin. Anthony Stokes also came close to extending the lead minutes later, but once again, the crossbar kept Cliftonville in the game.
22-year old Forrest caused the hosts endless problems throughout the evening and he sealed the win for Celtic with five minutes remaining as he ran at the Cliftonville defence at speed before firing a low strike past Devlin at his near post.
The victory gives Neil Lennon's men one foot in the third round of qualifying as they look to build on last season's impressive run in the competition, with the return game at Celtic Park taking place next Tuesday.