Cluj took the lead in the ninth minute through Florin Costea and substitute Patelis Kapetanos added a second goal for the Romanian side in the 65th minute.
Georgios Samaras came off the bench to pull a goal back for the Hoops with eight minutes remaining but the Scottish champions, who lost 1-0 to Ukrainian side Sevastopol earlier in the week, were unable to find an equaliser.
Celtic, warming up for their Champions League qualifying ties against Cliftonville later this month, were below full strength against a side further ahead in their pre-season preparations and they made a slow start.
Costea's well-taken opener - a curling shot over keeper Fraser Forster which went in off the post - woke the Parkhead side up and Gary Hooper drove a 20-yard effort wide.
Cluj quickly reasserted themselves and Forster did well to keep out Ciprian Deac's free-kick before Costea headed over.
Neil Lennon made a number of changes at half-time and Samaras replaced Amido Balde early in the second half as the substitutions continued from both sides.
Beram Kayal, who had taken over as skipper from Mikael Lustig at the interval, made a well-timed challenge to foil Paul Batin, but Batin then evaded Kelvin Wilson before setting up Kapetanos to steer the ball past Forster from 12 yards.
Defender Virgil Van Dijk spurned a good chance to pull a goal back for Celtic in the 73rd minute when he headed over from 10 yards, but Celtic did reduce the deficit in the 82nd minute as James Forrest's cross went in off Samaras' knee.