Roared on by home fans in their 50,000 capacity stadium on the Asian side of Istanbul, Fenerbahce produced a stunning performance capped by a brilliant Moussa Sow bicycle-kick goal and a Jeremain Lens free-kick.
Wayne Rooney managed to pull a goal back to create a tense final few minutes but it did nothing to brighten a miserable night for United.
United now face a tense three-way battle for qualification from Group A with Fenerbahce and Feyenoord who they now trail by one point after four games played.
Mourinho said that the team had been caught "sleeping" and had failed to heed his warnings over the toughness of playing away matches in Istanbul.
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He admitted the game had also exposed the lack of proper central defenders in the side and the lack of confidence in front of goal.
"What disappointed me more is I have quite a big experience of playing against Turkish teams," said the former Real Madrid and Chelsea boss.
"But we started the game sleeping. They start at 100mph; we start really slow. They go immediately in there, we are sleeping we don't react and then the game is a different story."
- 'Need stronger mentality' -
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"It is our fault that we let them be in front. We didn't deserve (to win)."
Mourinho also said that United were lacking confidence in front of goal and "one of the good things today is that Rooney finally scored a goal" that could boost him.
United also suffered a major blow with Paul Pogba limping off injured and Mourinho lamented that no yellow card had been given for the challenge on the French star.
"I only know that that the first two fouls (on Fenerbahce players) got two yellow cards... The foul on Pogba that brought him out of the game was not a yellow card."
Mourinho said he did not know yet the extent of the injury to Pogba.
"He is playing well for the team, he's creating but is missing a lot of goals."
Mourinho still believes United would still go through to the knock out stages, even though the team is now in third behind Feyenoord and Fenerbahce who both have seven points.
"We depend only on us. I think we are going to go through."
Mourinho said there would be no place for Bastian Schweinsteiger in the line-up against Swansea, even though the German international has now returned to first team training.