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.
Qualification for the next stage remains in the Reds' hands, but they will need to be more incisive against their next opponents.