Media playback is not supported on this device
United went ahead in the last-16 second leg tie before Nani was controversially dismissed for a high challenge.
Cristiano Ronaldo, on his return to Old Trafford, then tapped in the winner after Luka Modric's stunning equaliser.
"Independent of the decision, the best team lost. We didn't deserve to win but football is like this," Mourinho said.
Following a 1-1 draw in Spain, United had edged a tight second-leg opening in which captain Nemanja Vidic struck the post with a powerful header.
They finally took the lead - on the night and on aggregate - at the start of the second half, when Sergio Ramos deflected Nani's cross into his own net.
The match then turned in Real's favour just before the hour mark when referee Cuneyt Cakir dismissed the Portuguese for a high boot that made contact with Alvaro Arbeloa's rib-cage.
Mourinho's side took advantage of the decision in clinical fashion. Former Tottenham player Modric curled in a sublime equaliser after he came off the substitute's bench, and Ronaldo scored from close range after a stylish move.
"My feeling is that Manchester United were playing very well, were very compact and aggressive in a good way," Mourinho added. "I doubt that 11 v 11 we win the match.
"I know Manchester United are giants, not just physically but mentally. I know they have a manager who can motivate then, but I was waiting for us to play in a different way."
Before kick-off, Ronaldo was given a warm reception by supporters at the club where he had played a pivotal role in the winning of the Champions League in 2008, as well as three Premier League titles, in a six-year spell in England.
The attacker - who moved to Real for a record £80m fee in 2009 - admitted he had been overwhelmed by the reaction.
"It was unbelievable," he said. "It was an emotional moment for me, for both games, but especially tonight.
"I do not have words to explain how I feel. The supporters made me feel shy. I did not play how I play all the time."
When asked to comment on rumours of a potential return to Old Trafford, he replied: "The future I never know, but the supporters here and the club are massive.
"At this moment I am very happy in Madrid and I want to be there."
Real Madrid manager Jose Mourinho has claimed "the best team lost" after 10-man Manchester United were knocked out of the Champions League.