Without a victory in four Premier League games, United desperately needed a morale-lifting result in Turkey but were undone by a brilliant second-minute overhead kick from Sow and a superb curling set piece by Lens before Wayne Rooney smashed home a consolation.
While Mourinho's men are third in Group A, Zenit St Petersburg, Ajax Amsterdam, Shakhtar Donetsk and Schalke 04 have all sauntered through to the knockout rounds with two matches to spare.
What Mourinho would give for an in-form striker like Athletic Bilbao's Aritz Aduriz who became the first player to score five times in a Europa League match, his haul comprising three penalties in a 5-3 triumph over Racing Genk in Group F.
A gloomy pall has descended over Mourinho's reign at Old Trafford with the manager under scrutiny for his team's poor start and the disciplinary problems that have earned him a domestic one-match touchline ban.
After victories in their last two group games, Europe's second-tier competition had supplied United with some relief from their league struggles but they were sliced apart by Fenerbahce who exposed some glaring defensive weaknesses.
The two spectacular goals they conceded, however, were almost unstoppable.
The match had barely begun when Sow flung himself on the edge of the box into an overhead volley from a floated cross, smashing the ball past United keeper David De Gea and into the top corner.
Fenerbahce then took a stranglehold on the tie when Lens stepped up to nonchalantly curl a free kick around the wall and into the corner 14 minutes after the restart, leaving De Gea rooted to the spot as the ball bulged the net.
SUMMER FRIENDLY
It should have got a lot worse for United when second-half substitute Emmanuel Emenike threatened three times to extend the lead before Rooney drove the ball home from 25 metres to set up a nervy finish.
It was United's start, however, that most displeased Mourinho who railed at the lack of intensity in their play.
"Our problems started in our global attitude. They were playing the Champions League final, we were playing a summer friendly," he told BT Sport.
United, who suffered an injury blow when the world's most expensive player Paul Pogba was forced off in the first half, are now a point behind group leaders Feyenoord and Fenerbahce with two games remaining.
Zenit defeated Irish side Dundalk to maintain their perfect record in Group D while Shakhtar hit five for the second time in Group H as they overcame Belgians Gent 5-3.
Ajax reached the last 32 with a 3-2 Group G win over Celta Vigo while Schalke won Group I after goals from Junior Caicara and Nabil Bentaleb secured a 2-0 victory over Krasnodar and a fourth straight triumph.
Southampton, playing in the third tier of English football six years ago, came from a goal down to beat former European royalty Inter Milan 2-1 in a tempestuous encounter in Group K.
Mauro Icardi gave Inter the lead with a well-struck finish before tempers flared when Southampton were awarded a penalty at the end of the first half.
Dusan Tadic missed from the spot with the last kick of the opening 45 minutes but Virgil van Dijk levelled from close range and Yuto Nagatomo's own goal gave the hosts all three points.
Anderlecht were on the goal trail in Group C as Nicolae Stanciu and Lukasz Teodorczyk netted twice for the Belgians in their 6-1 rout of Germans Mainz 05.
AS Roma's Edin Dzeko bagged a double as they moved top of Group E after a 4-2 win at Austria Vienna.
(Editing by Tony Jimenez)
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