Striker Moussa Sow put the hosts ahead after 65 seconds with an exquisite overhead effort from Ali Kildirim's cross before Sunderland loanee Jeremain Lens curled in a brilliant set piece after the break.
The visitors dominated possession but did not threaten enough and saw Paul Pogba go off with a leg injury on the half hour mark.
Wayne Rooney silenced his critics with a superb 25 yarder late on but it proved a mere consolation in the infamous cauldron of the Sukru Saracoglu.
Our last victory on the road in Europe came in 2013 under David Moyes- 5-0 in Leverkusen- but qualification to the knockout stage remains in our hands with two games left (Feyenoord at home and Zorya away).
After the early setback of going behind, United went close through Marcus Rashford while Simon Kjaer cleared under pressure from the onrushing Rooney.
Lens headed wide at the other end and Fener keeper Volkan Demirel blocked bravely at the feet of out of form Zlatan shortly after he had come on from the stricken Pogba.
United pushed to restore parity and went close twice in quick succession through Rooney, who headed wide, and half-time substitute Juan Mata who lashed an effort narrowly off target.
Just when an equaliser looked imminent, however, Fenerbahce doubled their lead with another superb goal on the hour mark.
Ibrahimovic's foul on Mehmet Topal 25 yards out was punished by Lens who smashed home an unstoppable free-kick that left David de Gea stranded.
Henrikh Mkhitaryan came on for his first appearance in nearly two months but Fener continued to threaten when de Gea's point blank block kept out substitute Emmanuel Emenike.
De Gea was forced into another excellent stop from the same player and then breathed a sigh of relief when Topal's cross narrowly evaded the onrushing Nigerian.
Rooney gave the Reds a late lifeline with a wonderful long range finish and then had a header cleared off the line from a Mata corner in injury time.
The five consecutive European away defeats (Wolfsburg, Midtjylland, Liverpool, Feyenoord and Fener) extends our run to an unwanted club record seven matches without a win on the road in European competition.
Feyenoord and Zorya drew 1-1 in Ukraine which means that the Dutch side lead the way along with Fener on seven points with United a further point behind in Group A.
United Faithful Man of the Match: Wayne Rooney