Needing to avoid defeat to secure second place in Group E ahead of Lazio, Cluj ultimately had a comfortable night against a Celtic side who had already qualified and were chasing successive away victories in Europe for the first time since 2003.
Abdel Billel Omrani and Lacina Traore passed up chances but defender Andrei Burca made no such mistake to head the hosts in front in the 48th minute.
Celtic could not kick-start an underwhelming display, and Cluj secured their passage into the round of 32 for the first time when Damjan Djokovic prodded home with 20 minutes remaining.
Fraser Forster has been Celtic's first-choice goalkeeper in Europe this season, but stand-in Craig Gordon showed his quality with a fine save to deny Omrani in the 18th minute.
Gordon would have had little chance of preventing Traore opening the scoring, only for the former Monaco striker to send his close-range header over.
But Celtic did not get away with further sloppy defending three minutes after the break – Burca heading home unmarked from Ciprian Deac's inswinging corner.
Deac almost went from provider to scorer three minutes later, Gordon pulling off another top-quality stop after a defensive mix-up left Celtic exposed.
Leigh Griffiths' header forced keeper Giedrius Arlauskis into action at the other end, yet Celtic's hopes of a comeback were ended when Djokovic turned in Deac's cut-back to ensure Cluj's progression.