Maccabi came into the second-round second-leg duel down 1-0, and took an early lead on a goal by Nick Blackman in the 15th minute.
However, Cluj answered with a pair of first-half tallies and even a Yonatan Cohen 48th-minute marker was not enough to keep the yellow-and-blue from falling 3-2 on aggregate.
Tel Aviv will now enter the third round of Europa League qualifying and likely face Sudova from Lithuania.