Florin Costea opened the scoring for the Romanian side in the ninth minute with substitute Patelis Kapetanos doubling that lead in the 65th minute.
Hoops substitute Georgios Samaras pulled a goal back in the 82nd minute but it was not enough to prevent the Scottish champions losing their second challenge match this week, following the 1-0 loss to Ukrainian side Sevastopol.
However, Celtic manager Neil Lennon should not be unduly worried. Their opponents are further ahead in their pre-season preparations and at times it showed.
Celtic eyes are all on the Champions League qualifying ties against Northern Irish side Cliftonville later this month and to that end, the former Celtic skipper is assessing his squad.
A far from full-strength Parkhead side, captained by defender Mikael Lustig and with Calum Waters making his first start for the senior side, found themselves a goal behind during a shaky start when Costea's well-taken shot from inside the box curled high past keeper Fraser Forster and in off the post.
The goal shook Celtic out of their slumber and moments later striker Gary Hooper drove wide of the target from 20 yards to end a swift counter.
The Romanians, who had former St Johnstone striker Gregory Tade in their attack, reasserted themselves and Forster had to make a great save from Ciprian Deac's whipped-in free-kick before Costea sent a header from a corner high over the bar.
The Celtic keeper made two further saves before the break as the Parkhead side were kept at arm's length.
James Forrest, Darnell Fisher and Charlie Mulgrew replaced Dylan McGeouch, Lustig and Liam Henderson respectively for the start of the second-half with Beram Kayal taking the armband and soon Samaras replaced Amido Balde as the substitutions on both sides continued.
Celtic began to look more purposeful but just after the hour mark, after giving the ball away, Kayal had to make a last-gasp tackle at the expense of a corner to prevent Paul Batin getting a shot away inside the box.
The corner came to nothing but Batin was less hesitant minutes later when he evaded Parkhead defender Kelvin Wilson to set up Kapetanos to steer the ball past the helpless Forster from 12 yards.
Cluj looked in control and more likely to add to their lead but in the 73rd minute Celtic defender Virgil Van Dijk missed a good chance by heading a Mulgrew corner over the bar from 10 yards.
The Parkhead men reduced the lead with eight minutes to go when Forrest's cross from the right came off the knee of Samaras with Mulgrew moving quickly out of the way to allow the ball to cross the line.