United beat Fenerbahce 4-1 at Old Trafford a fortnight ago, but arrived in Turkey after drawing with Burnley 0-0, during which it dominated and had 37 chances.
Again, Jose Mourinho's side failed to convert a glut of possession into goals, but this time it also struggled to create opportunities. Its attacking intent increased briefly after Zlatan Ibrahimovic replaced the injured Paul Pogba half an hour in.
United's struggles in front of goal, however, are being reflected by Ibrahimovic, who netted five goals in as many matches to get his United career off to a flying start, but has managed to score just once in the last 11 matches.
"They deserved to win. Football is not just about quality, it's about effort, commitment, playing the limits, and giving everything," Mourinho said. "A team that concedes a goal after two minutes is a team that is not ready, is a team that is not mentally prepared, not focused, not concentrated ...
"After their second goal we had a reaction ... but we didn't deserve more. Our problem started in our attitude. They were playing the Champions League final and we were playing a summer friendly."
United's David De Gea was by far the busier goalkeeper.
Moussa Sow put Fenerbahce in front after only 66 seconds with a stunning overhead kick from just inside the area, and Jermaine Lens doubled the Turkish side's lead with a magnificent free kick in the 59th, much to the delight of a raucous home crowd. De Gea hardly moved for it.
Wayne Rooney netted a late consolation, smashing the ball in from 30 yards to move level with Ruud van Nistelrooy on 38 European goals for United. The captain is two goals from tying Bobby Charlton's record of 249 for United.
Fenerbahce leapfrogged United into top spot in Group A, level on points with Feyenoord, which drew at Zorya Luhansk 1-1. United slipped to third, one point off the pace.