Replays appeared to show that the creator of the decisive moment- Nicolai Jorgensen- was in an offside position but play was allowed to continue and Vilhena fired home with eleven minutes remaining.
United manager Jose Mourinho made eight changes to the line-up with David De Gea, Eric Bailly and Paul Pogba the only players to keep their places from the derby defeat.
Early Eredivisie leaders Feyenoord came into the match in fine form, boasting a 100% record so far and, with chances at a premium at the De Kuip, the hosts went closest early on.
De Gea saved well from Steven Berghuis while Anthony Martial went close on 24 minutes when he fizzed a shot wide after neat build up play from the lively Marcus Rashford.
The Reds breathed a sigh of relief at the other end five minutes before the break when a period of pressure from Feyenoord saw Jens Toomstra lash an effort over the bar when well placed.
Ander Herrera- who had provided energy and thrust against City- powered forward immediately after the restart and chanced his luck from distance but his shot flew wide.
Thereafter the pattern of the game continued with both sides enjoying decent periods of possession without any real clear cut openings.
With Jose keen to get off to a winning start in the group, a triple change just beyond the hour mark saw Memphis, Zlatan and Ashley Young introduced in place of Rashford, Martial and Juan Mata.
That failed to spark the visitors into life, though, and it was the hosts who made the decisive breakthrough on 79 minutes- albeit in a contentious manner.
Former Aston Villa man Karim El Ahmadi picked out the run of Jorgensen who raced clear on the right as the flag stayed down and his cross fell to Vilhena who made no mistake with his finish.
Substitute Young saw a shot scrambled wide and an excellent Ibrahimovic free-kick was superbly saved by former Liverpool keeper Brad Jones as the Dutch side held out.
A disappointing start in Europe but there's no need for panic after one game with plenty of time to recover in a group that we remain favourites to get out of.
Zorya- United's next opponents in the Europa League- produced a creditable performance to draw 1-1 at home to Fenerbahce in the other match.
United Faithful man of the match: Not many to choose from but Ander Herrera stood out on a disappointing night.