The Scottish champions defeated HJK Helsinki 4-1 across two legs to reach the play-offs and learned their fate at UEFA's headquarters in Nyon on Friday morning.
Helsingborgs are the first Swedish team that Celtic have ever faced in European competition.
The Bhoys will play away from home in the first leg, which will take place on August 22 or 23. The second leg at Parkhead will take place on August 28 or 29.
The two clubs have some common ground as both played host to the extravagant talents of Henrik Larsson.
The former Sweden international played for Helsingborgs from 1992 to 1993 and 2006 to 2009, as well as enjoying seven glorious years in Glasgow between 1997 and 2004.
Troubled Malaga will have to beat Champions League regulars Panathinaikos if they are to reach the group stage for the first time.
Borussia Moenchengladbach, also hoping to qualify for the first time, have been paired with Ukraine's Dynamo Kiev, who have reached the semi-finals three times.
Italy's Udinese, beaten by Arsenal at the same stage last season, will face Braga of Portugal. Defeat would leave Italy with only two teams in the group stage.
Malaga finished fourth in La Liga last season but their outlook is uncertain after they announced "internal restructuring" amid reports that Qatari owner Sheikh Abdullah Bin Nassar Al-Thani wants to sell up after only two years.
In the last week, midfielder Santi Cazorla has joined Arsenal while Venezuela forward Salomon Rondon has moved to Russia's Rubin Kazan.
Their Greek opponents will be attempting to reach the group stage for the 10th time.
In other ties, Spartak Moscow will face Fenerbahce, who were excluded from the competition last season over a Turkish match-fixing scandal, and Lille will meet Copenhagen.
Cypriot champions AEL Limassol, hoping to emulate compatriots APOEL Nicosia who reached the quarter-finals last season, must first beat Belgian champions Anderlecht.
Celtic, the only former European champions among the teams in Friday's draw, were pitted against Swedish champions Helsingborg.
There is no English side in the playoffs because Chelsea won the Champions League last term and therefore go straight into the group stage despite having finished outside the Premier League's top four.
Full Champions League play-off draw:
FC Basel (SUI) v CFR Cluj (ROU)
Helsingborgs (SWE) v Celtic (SCO)
BATE Borisov (BLR) v Hapoel Kiryat Shmona (ISR)
AEL Limassol (CYP) v Anderlecht (BEL)
Dinamo Zagreb (CRO) v NK Maribor (SVN)
Braga (POR) v Udinese (ITA)
Spartak Moscow (RUS) v Fenerbahce (TUR)
Malaga (ESP) v Panathinaikos (GRE)
Borussia Moenchengladbach (GER) v Dynamo Kiev (UKR)
Lille (FRA) v Copenhagen (DEN)