Goals from Mikael Lustig, Georgios Samaras, and James Forrest helped Neil Lennon's side ease past their Northern Irish opponents and take a healthy aggregate lead
Mikael Lustig and Georgios Samaras were on target for the visitors, with James Forrest adding to the tally late on.
The Bhoys started brightly at Solitude, with Kris Commons rattling the crossbar after just three minutes. Commons' header beat goalkeeper Conor Devlin, but bounced clear after a great cross in from James Forrest.
Celtic controlled the game for much of the half, and forced George McMullen into a mistake which nearly ended up with the midfielder putting the ball through his own net.
The resulting corner saw the Scots go ahead. Devlin came out to punch, but Lustig got to the ball first and powered his header into the net. Celtic added a second on the half-hour mark as Samaras swivelled in the box to drive the ball past Devlin after a neat ball in from Commons.
Commons hit the woodwork again just before half-time, seeing his effort rebound off the post with Devlin well beaten. After the break, Liam Boyce had Cliftonville's first real chance, but was denied a goalscoring opportunity when Efe Ambrose got across to block.
A weak header from Lustig allowed Stephen Garrett in on goal, but the forward could not beat Fraser Forster, who had a quiet evening in the away goal. Irish striker Anthony Stokes had a couple of chances of his own, but his shooting let him down with Devlin more than equal to his efforts.
Forrest used his pace to bag the visitors' third with a fine solo goal, but Cliftonville keeper Devlin should have done better to stop the forward's effort. Neil Lennon's side now take a big lead back to Celtic Park ahead of the second leg on Tuesday, July 23.