A superb second-half strike from the visiting side was enough to hand it a famous victory at Old Trafford as Sir Alex Ferguson's men slumped to defeat in their final group match.
CFR Cluj, meanwhile, arrived at Old Trafford safe in the knowledge that the club could finish no lower than third in the group, and the Romanians comfortably held their own in a first half which saw chances for both sides but no goals.
Cluj had the best of the early openings. Midway through the first half Madou Sagou was put clean through on goal but was denied by a superb challenge from Wotton who slid in at the crucial moment to block his shot and force the corner.
Minutes later and the action switched to the other end as Tom Cleverley had an excellent chance to open the scoring. With Danny Welbeck closing down the Cluj defense and managing to nick the ball away, the possession switched to the midfielder but his heavy first touch allowed goalkeeper Mario Felgueiras the chance to come smartly off his line and block his shot.
With halftime looming, United improved and carved out another excellent opportunity. After Felgueiras had saved well from Welbeck and Rooney, United kept the momentum alive and found Smalling from a corner but he could not direct his header on target from close range as the first half ended goalless.
CFR Cluj began the second half in determined fashion and took the lead after just 10 minutes.
Substitute Paul Scholes, who replaced Cleverley at the end of the first half, gave the ball away and it eventually fell to Luis Alberto. The midfielder was allowed to take a few touches before unleashing an unstoppable shot which flew into the top corner from 25 yards.
With United struggling to work Felgueiras in the CFR Cluj goal, Sir Alex brought Federico Macheda on for Nick Powell with 20 minutes remaining to add some attacking impetus.
Yet for all United's efforts the team found it tough going against a CFR Cluj side content to hold the lead. Felgueiras in the visitor's goal in particular was in commanding form, alive to any danger, and always quick off his line to deny any potential goalscoring opportunities.
While the result mattered little to United, it did to CFR Cluj which despite recording a famous win at Old Trafford still exit the competition. A win for Galatasaray in Braga in the other game in the group, enough to relegate the Romanians to the Europa League.
Manchester United, meanwhile, heads through as the group winner but will now turn its attention back to domestic matters and the small matter of a trip to Etihad Stadium to face Manchester City.