The Hoops had already won Group E to secure a last-32 spot before the trip to Romania and manager Neil Lennon left a clutch of key players in Glasgow with 18-year-old midfielder Scott Robertson making his debut.
Following an untidy first half, defender Andrei Burca headed in from a corner in the 48th minute and midfielder Damjan Djokovic added a second in the 70th minute to secure second spot for Dan Petrescu's side.
The Scottish champions had won four and drawn one of their previous five section fixtures and it was their first defeat in 13 matches in all competitions.
Lennon rests seven, one suspended
However, despite the performance, Celtic, with their stars back, can look forward to Monday's draw with confidence.
Much of the attention around the game was on which players Lennon rested.
Fraser Forster, Scott Brown, Callum McGregor, Odsonne Edouard, Tom Rogic and Ryan Christie, as well as suspended skipper Scott Brown and 19-year-old right-back Jeremie Frimpong, not registered for European football, were all missing.
Indeed, the Scottish champions showed nine changes to the side which started in the Betfred Cup final win over Rangers at Hampden Park on Sunday, with midfielder Olivier Ntcham named captain.
Unsurprisingly, the visitors looked out of sorts in the early stages, although Cluj were little better with both sides guilty of sloppiness.
In the 18th minute Hoops' 36-year-old goalkeeper Craig Gordon, starting for the first time since a Betfred Cup tie against Partick Thistle in September, saved Billel Omrani's strike from inside the box after the Cluj attacker was set up by Lacina Traore.
Quality at a premium
Celtic's first effort on goal came in the 32nd minute when defender Nir Bitton fired in a long-range strike but Cluj keeper Giedrius Arlauskis was not unduly troubled.
Quality remained at a premium, exemplified when Traore, enjoying a battle with Parkhead stopper Christopher Jullien, headed a free-kick from Ciprian Deac past the far post.
Just before the break a 25-yard drive from Celtic playmaker Mikey Johnston was deflected wide for a corner but, like most of the action in the first half, that came to nothing.
Defender Kristoffer Ajer replaced Jullien for the start of the second half but Traore continued to cause trouble and, after his shot had been deflected behind for a corner, Deac zipped it in from the right and Burca escaped from Boli Bolingoli to power a header in off the near post.
Moments later, Gordon prevented Deac from doubling the home side's lead with another good diving save as the Hoops defence opened up again.
In the 57th minute Parkhead striker Leigh Griffiths, making his first start since August, headed a Moritz Bauer cross straight at Arlauskis before the keeper gathered a 30-yard strike from Ntcham.
Dembele makes history
Vakoun Issouf Bayo came on for Lewis Morgan in the 67th minute but three minutes later, following a terrific cross from Emmanuel Culio, Deac cut the ball back for Djokovic to hammer it high into the net.
Celtic's 16-year-old attacker Karamoko Dembele quickly replaced Johnston to become the youngest ever player for the club in a European competition but the home side retained their threat and Gordon made further saves from Deac and Traore, with the Parkhead side struggling to the end.