Striker Moussa Sow gave Fenerbahce the lead within 65 seconds with an overhead kick which sailed over David de Gea, and Jeremain Lens curled in a brilliant free-kick in the second half.
The visitors dominated possession but saw world-record signing Paul Pogba go off with a leg injury in the first half.
Captain Wayne Rooney smashed in a 25-yard consolation goal late on.
Defeat means United are third in Group A – one point behind leaders Feyenoord and Fenerbahce – with two games left, against the Dutch side at Old Trafford and away to Ukraine’s Zorya Luhansk.
Mourinho slams players
Jose Mourinho felt let down by Manchester United’s players, accusing them of ignoring his advice and treating the Europa League clash at Fenerbahce like a “summer friendly”.
“They (Fenerbahce) were playing the Champions League final and we were playing a summer friendly. That is the reality of the way the game started.”
“What disappointed me more is that I have quite a big experience of playing against Turkish teams,” Mourinho, who lost Paul Pogba to injury after 30 minutes, said in the post-match press conference.
“I played twice with Real Madrid and twice with Chelsea, I played with Porto too, so I pass to the players all my experience.
And my experience is that when you play against them in our home, it’s easy. It’s always 2-0, 3-0, 4-0, it’s easy.
“You come to Turkey and it is completely different story, so I told them to wait for that, told them not to be focused on the easy game we had at Old Trafford and to be ready for a completely different game,
“We started the game sleeping. I know that we have a problem and especially away from home, it’s difficult to play without two proper central defenders.
“Football play starts with goalkeeper, two strong central defenders. We are playing with two left-backs (Daley Blind and Marcos Rojo) as central defenders and poor Blind is even having to play central defender on the right, which is even more complicated.
“So I think at this moment we lack that structure there, but they start 100 miles per hour and we start really slowly.
“They had the kick-off, they go immediately, the referee gives a free-kick. They take the free-kick in a fast way, we are sleeping, we don’t react. Then it’s a ball in the other side, we lose the second ball. Then the cross is coming, we don’t mark Sow in the box and then the game is a different story.
“Then they are where they want to be. They are in front, then they can control the game, they can defend with a lot of bodies, they can wait for counter-attacks. They can simulate a few fouls, put some pressure on the referee, stop the game, win time, provoke some conflicts.
“But it is not their fault, it our fault that we let them be 1-0 in front.
“Then obviously when you are losing 2-0, the players had a reaction and they tried and probably they could (have equalised).