Neil Lennon's side will play the first leg of their second-round qualifier on July 16/17 before hosting the return game a week later.
Celtic reached the knockout stage of last season's competition thanks in part to a group-stage victory over Barcelona in Glasgow, and striker Anthony Stokes is looking forward to this term's opener.
He told the club website: "It's a great draw for both clubs "Cliftonville had an exceptional season and it will be a tough game and we will have to be at our best, but we're going to have a good pre-season and will be ready.
"Both atmospheres will be electric. I'm sure they will have a good crowd over here for the game as I know there is a decent relationship between the two clubs.
"I know a lot of their fans would be Celtic fans as well or would look for us to do well, though maybe not in these two games."
Elsewhere, Welsh club the New Saints were paired with Legia Warsaw of Poland, while Irish champions Sligo Rovers will take on Ole Gunnar Solskjaer's Norwegian side Molde.
The winners of the second-round ties must come through two further rounds of qualifying to make the group stage, although should Celtic progress they will be seeded for any further qualifying rounds.
UEFA Champions League first and second qualifying round draws:
First qualifying round:
Shirak v Tre Penne
Lusitanos v EB/Streymur
Ties to be played 2/3 and 9/10 July
Second qualifying round:
Neftci v Skenderbeu
Steaua Bucuresti v Vardar
Plzen v Zeljeznicar
FC Sheriff v Sutjeska
Birkirkara v Maribor
Sligo Rovers v Molde
IF Elfsborg v FC Daugava
HJK Helsinki v Nomme Kalju
FK Ekranas v FH Hafnarfjordur
The New Saints v Legia Warsaw
Celtic v Cliftonville
Dinamo Zagreb v CS Fola Esch
Gyor v Maccabi Tel-Aviv
BATE v Shakhtyor Karaganda
Shirak/Tre Penne v Partizan Belgrade
Slovan Bratislava v Ludogorets
Dinamo Tbilisi v Lusitanos/EB/Streymur
Ties to be played 16/17 and 23/24 July