Jose Mourinho made eight changes from the team beaten by Manchester City and an Anthony Martial shot going wide was their only first-half chance.
Zlatan Ibrahimovic came on for the visitors and headed wide as he failed to turn around his side's fortunes.
Instead, Nicolai Jorgensen, who was offside, crossed for Tonny Vilhena to drive in Feyenoord's late winner.
Ibrahimovic went close to an equaliser with a drilled free-kick but it was saved at the second attempt by former Liverpool keeper Brad Jones.
The defeat means United have lost four successive European away games for the first time.