Brazilian Luis Alberto hit an unstoppable swerving and dipping shot after 56 minutes to sink a second-string United but Galatasaray's 2-1 win at bottom side Braga put them into the last 16.
Cluj finished Group H level on 10 points with Galatasaray, behind winners United on 12, but the Romanians lost out on a worse head-to-head record and have the Europa League as consolation.
Steaua Bucharest, the 1986 winners, reached the final of Europe's elite club competition in 1989, but no Romanian club has reached the knockout stages of the Champions League since it was formed in the 1992-93 season.
Cluj, making their third Champions League appearance after group stage outings in 2010-11 and 2008-09, survived pressure from the home team in the first half when goalkeeper Mario Felgueiras denied Tom Cleverley and Wayne Rooney.
Felgueiras also repelled a Danny Welbeck header as United looked to grab the advantage.
However, it was Luis Alberto's moment of genius in the second half that stole the show for the visitors in a largely dour affair.
(Writing by Tom Pilcher in London, Editing by Ed Osmond)
