It was United's second loss in less than a week after getting beaten 1-2 by fierce local rivals Manchester City in the English Premier League last Saturday.
Feyenoord midfielder Tonny Trindade de Vilhena scored the winner in the Group A match, side-footing the ball past United goalkeeper David de Gea in the 79th minute here on Thursday.
United played without first-team regulars Wayne Rooney, Luke Shaw and Antonio Valencia but brought on star striker Zlatan Ibrahimovic in search of the equaliser but were left frustrated.
In Milan, Miguel Vitor's 54th minute strike and a 69th-minute free-kick from Maor Bar Buzaglo helped Hapoel beat Inter at the San Siro stadium in a Group K match.
It was Inter's new coach Frank De Boer's fourth match in charge, of which he has now won just one.
Southampton went to the top of Group K after two first-half goals from Charlie Austin and a Jay Rodriguez strike provided a 3-0 win against Sparta Prague.
In Group E, Italy's AS Roma were held to a 1-1 draw at Viktoria Plzen.
Also in Group E, Austria Vienna earned a win 3-2 against Astra Giurgiu.
