Brazilian Luis Alberto struck an unstoppable swerving shot after 56 minutes to condemn a second-string United to their first home loss in Europe's elite competition since 2009, but Galatasaray's 2-1 win at Braga put the Turkish side through.
Cluj finished Group H level on 10 points with Galatasaray, behind winners United on 12, but the Romanians lost out on 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, inflicting a first home defeat on United since they lost to Besiktas in November 2009.
"It's a fantastic goal they scored to beat us," United manager Alex Ferguson told the club's website of the superb strike past goalkeeper David de Gea.
"He bent it away from David, who had no chance with it. It was just one of those nights. We didn't take our chances," added Ferguson, unfazed by two straight Champions League defeats after his side were assured of qualification by winning their first four games.
"We qualified early and after that we experimented. That will hold us in good stead for the next phase," the Scot told Sky Sports, adding that midfielder Cleverley had picked up a calf injury at the end of the first half.
United's last-16 opponents will be known when the draw is made on Dec. 20, with Ferguson's men due to face one of the group runners-up.
Nine-times European champions Real Madrid, seven-times winners AC Milan, 1987 and 2004 champions Porto, twice finalists Valencia, 2009 UEFA Cup winners Shakhtar Donetsk or Scotland's Celtic await.
Cluj head into uncharted waters when they resume European competition in the Europa League last 32 in February, their first taste of continental action after Christmas, though coach Paulo Sergio said his team were upset not to have qualified.
"We played very, very well. Our game went as we had worked for and expected, and we did not allow Manchester United to impose their style of play," he told UEFA's website.
"The players are sad now, but they have to be proud, because it is not common to win at Old Trafford," added Sergio, who replaced Ioan Andone in October, midway through the Romanian champions' group-stage campaign.
