Alberto put the Romanian champions ahead just before the hour but with Galatasary coming from behind to win 2-1 in Braga, Cluj were eliminated on the head to head ruling.
Paulo Sergio's side, who must have believed a victory at Old Trafford would take them to the next stage, will now feature in the Europa League in the new year.
United were through even before the 1-0 defeat at Galatasary last month and progress as group winners.
But despite making 10 changes ahead of the Manchester derby with City on Sunday, Alex Ferguson will still be frustrated at his team's second successive European defeat.
It was his team's first home loss since a 1-0 defeat to Besiktas in November 2009.
But, having taken the unusual step of criticising the defending of his side, who have now conceded 33 goals this season, before the game, he will be angry to have witnessed more errors against Cluj.
Alberto's goal was the 15th occasion United have fallen behind this season and they have only kept four clean sheets in 23 matches in the current campaign.
Danny Welbeck went close to giving United an early breakthrough but his header from a Ryan Giggs cross flew wide of goal.
There was more evidence of what has been troubling Ferguson when a loose Giggs backpass put Chris Smalling under pressure and the England defender's poor clearance allowed Rui Pedro to shoot, only for David De Gea to make a smart save.
The Romanians pressed forward whenever they could and Pedro swapped passes with Modou Sogou before the striker's effort was deflected wide by United youngster Scott Wootton.
Camora found more space when he got in behind Phil Jones only to see his cross cleared wide by Tom Cleverley with Sogou about to pounce at the near post.
United's frustration continued when Cleverley picked out Wayne Rooney, who was the only survivor from Saturday's win at Reading, on the right but after his initial effort was blocked he lifted his follow-up into the stand.
The hosts threatened when Welbeck robbed a defender and sent Cleverley through only for goalkeeper Felgueiras to rush out to make a block.
Rooney was denied by the Cluj goalkeeper when he attempted a clever chip and after the Romanians failed to clear, Javier Hernandez clipped in a cross that was headed just wide by Smalling.
Cleverley limped off to be replaced just before the interval and United continued to struggle to establish dominance after the restart when Rooney thumped a free-kick straight into the wall.
10 minutes into the second period, Cluj secured the lead they craved when Paul Scholes, who had replaced Cleverley, gave the ball away on halfway.
The veteran then dived in to allow Alberto the space to beat De Gea with a thumping shot from 30 yards.
United rarely looked capable of finding a way back and Felgueiras had no problem in swatting away a Giggs free-kick from the right.
Even a rare substitute appearance from Federico Macheda failed to inspire United and Scholes had a late effort blocked.
Italian striker Macheda also had an header hacked off the line as United limped to a second successive Champions League loss.