Memorable strikes from Moussa Sow and Jeremain Lens put the hosts in control before Dick Advocaat's side were able to hold on for a victory despite Wayne Rooney giving his side hope with an 89th-minute goal of his own.
The result sees Fenerbahce climb top of Group A with just two games remaining, while the Red Devils drop down to third and are now a point behind the top two.
Manchester United's team selection made the headlines as Jose Mourinho dropped Zlatan Ibrahimovic to the bench in favour of Rooney, but it was Fenerbahce's surprise starter who stole the show early on with a spectacular opening goal after just 69 seconds - the fastest goal conceded by United in Europe since November 2002.
The goal ensured that United have now conceded in seven consecutive European away games for the first time since 1965, and left Mourinho's side staring down the barrel of an eighth defeat in their last nine European games on the road.
Things got even worse for the visitors when Paul Pogba found himself on the end of a heavy challenge from Josef Souza and, despite trying to continue, the world's most expensive player was eventually forced off and replaced by Ibrahimovic.
Before his departure, though, the France international fired a tame effort into the arms of the keeper having burst into shooting space, while Rooney also failed to take a well-disguised Anthony Martial pass into his path when a better touch would have left him with only the keeper to beat.
Fenerbahce responded with a chance of their own moments later as Volkan Sen clipped a ball into the box, but Lens could not get enough on his glancing header having darted in front of Luke Shaw to get to the cross.
Marcus Rashford was the next to come close shortly after the half-hour mark when he collected a pass from Ibrahimovic before ducking inside and firing a cross/shot across goal, but Volkan Demirel got down to make the stop.
The hosts continued to pose a threat on the break despite conceding the vast majority of possession, and they almost doubled their advantage in first-half stoppage time when Alper Potuk's whipped delivery was nodded over by Sow at the back post.
The half-time whistle ensured that United have now failed to score a first-half goal in seven of their last eight matches in all competitions, and Mourinho responded with another change at half time as Juan Mata replaced Morgan Schneiderlin.
Mata had a half-chance to level things up 12 minutes after the restart too, but he snatched at his effort and dragged it wide of the far post having been teed up by Rashford.
It was Fener who got the all-important second goal shortly before the hour mark, though, and it was another memorable strike as Lens curled a 25-yard free kick past a statuesque De Gea after Ibrahimovic had been penalised for a foul on Mehmet Topal.
Things could have been even worse for United too when the hosts cut right through the defence with a rapid counter-attack to send Emmanuel Emenike through on goal just seconds after his introduction from the bench, but the former West Ham United striker could not take the ball in his stride and was eventually denied by a mixture of Daley Blind and De Gea.
Emenike continued to look the most likely on the break as Fenerbahce went in search of a killer third goal, but he was once again kept out by De Gea before falling agonisingly short of tapping home a low cross from Lens at the far post.
Those missed opportunities almost came back to haunt Advocaat's side when Rooney drilled a 25-yard piledriver into the top corner with just one minute left of normal time, moving within two goals of Sir Bobby Charlton in United's all-time goalscoring list and level with Ruud van Nistelrooy as the club's top scorer in European competition with 38.
Mourinho's side subsequently piled on the pressure throughout the four minutes of added time, and Rooney almost made it a dramatic late double with a header that needed to be nodded off the line by Ismail Koybasi.
The hosts were able to hold out for three vital points that see them take control of Group A, though, while also making it six European home wins in a row and eight without defeat.
United, meanwhile, are still winless away from home in Europe since November 2013, while in all competitions they have now won just two of their last seven games.