The visitors looked flat against a razor-sharp performance by the Turkish side
The visitors looked flat against a razor-sharp performance by the Turkish side, who took the lead early on with a staggering Moussa Sow overhead kick and added another with a Jeremain Lens free-kick in the second half.
A beautifully-judged long-range strike by Wayne Rooney late on led to a tense finale but could barely salvage United's pride.
The defeat increased the pressure on Mourinho, whose side have been struggling for form in the Premier League and now sit third in Europa League Group A with two games to go after rivals Feyenoord drew 1-1 with Zorya Luhansk in Ukraine.
"It was like they were playing a Champions League final and we were playing a summer friendly," Mourinho admitted at full-time.
He said he had been "disappointed" United had not taken on board his advice that that the away leg in Istanbul would be very different to the easy victory the team enjoyed against the Turkish team at home.
"I passed all my experience. Play against them in our home is easy... you come to Turkey and it's completely different.
"But we started the game sleeping. They start at 100mph -- we start really slow."
United enjoyed an easy 4-1 victory when the teams met at Old Trafford two weeks ago but they were thrust onto the back foot here inside two minutes as Fenerbahce went in front.
Sow met a cross from the left with a spectacular overhead kick that arced over a helpless David de Gea in the United goal and created an immediate cauldron of noise in the 50,000-capacity stadium.
United looked shell-shocked by the Senegalese striker's early stunner while Fenerbahce were roared on by their ecstatic home support.
Riding out deafening whistles from the home fans every time they touched the ball, United tried to press forward as Mourinho, hands in pockets, cut a distinctly unimpressed figure.
With Paul Pogba limping off injured in the 29th minute, Mourinho brought on Zlatan Ibrahimovic, but the home side continued to look more dangerous up front despite ex-United striker Robin van Persie being left on the bench.
Mourinho lamented that no yellow card had been given for the challenge on Pogba, saying he did not know the extent of the injury.
"I only know that that the first two fouls (on Fenerbahce players) got two yellow cards... the foul on Pogba that brought him out of the game was not a yellow card."
A scuffle with Simon Kjaer aside, Ibrahimovic struggled to have an impact on the game in an increasingly hostile atmosphere and it was a foul by the Swede which gave the hosts the chance to double their lead just before the hour mark.
Ibrahimovic was penalised 25 yards out and Dutchman Lens stepped up to send the home faithful into raptures with a stunning strike that again gave De Gea no chance.
Fenerbahce grew in confidence and substitute Emmanuel Emenike should have made it three but was blocked by a brilliant De Gea reflex save when one on one with the 'keeper. De Gea also tipped away a long-range effort from Emenike in the 75th minute.
Rooney's superb long-range strike that arrowed past Volkan Demirel gave United brief hope but he then had a header cleared off the line as Fenerbahce held on for a famous win.
"I think the team deserve credit for the way they played as a team," said coach Dick Advocaat.
"When we had the goal we were not scared to attack. We scored two but could have had four or five."
Mourinho also said "one of the good things today is that Rooney finally scored a goal" that could boost him going forwards.