A drab 90 minutes saw both teams struggle to create opportunities, but Man United lost a second successive match after Vilhena found the bottom corner with a smart finish in the 79th minute of the Group A fixture.
Marcus Rashford came into the team at centre-forward, while Matteo Darmian, Chris Smalling, Marcos Rojo, Morgan Schneiderlin, Ander Herrera, Juan Mata and Anthony Martial were all given chances to impress as Luke Shaw, Antonio Valencia, Wayne Rooney and Henrikh Mkhitaryan were left at home.
As for Feyenoord, former Liverpool forward Dirk Kuyt started in central midfield, while Australian goalkeeper Brad Jones, who represented Liverpool between 2010-2015, started his first European game for the Dutch outfit.
It was a cagey opening 10 minutes in Rotterdam as both teams battled for early possession, but it was Man United that were looking the more threatening with the pace of Rashford and Martial in the final third of the field.
The game's first shot on target came from Feyenoord in the 11th minute when Steven Berghuis struck one from the edge of the box, but De Gea was on hand to make the save, before Vilhena tested the Spanish goalkeeper from distance.
Man United's first attempt came in the 17th minute when Pogba, who was operating in a three-man midfield, sent a free kick towards the top corner, but Jones shuffled across to make the save as the score remained goalless after a tight opening to the clash in Netherlands.
The away side had another opportunity in the 23rd minute when Darmian found Martial in space just outside the Feyenoord box, but the Frenchman's low effort, which was arguably the best of the first period, just missed the post.
Feyenoord enjoyed the better of the 15-minute period before the end of the first half and had a super chance of their own to make the breakthrough in the 40th minute when Terence Kongolo set up Jens Toornstra at the far post, but the midfielder fired harmlessly wide under little pressure.
Down the other end, Rashford met a cross from Darmian as Man United looked to strike before the interval, but the teenager's header dropped wide of the post as the two teams headed down the tunnel on level terms.
Mourinho resisted the temptation to make any changes at the break and the visitors started the second period on the front foot, with Herrera striking wide of the post from distance, before Pogba hit one over Jones's crossbar three minutes later.
Man United remained the better team as they continued to press high up the field, but Feyenoord were still a threat and Karim El Ahmadi sent one wide of De Gea's post in the 57th minute of the Group A clash.
The home side again came close just past the hour when Vilhena struck one from the edge of the box, but his effort was always wide. That half-chance, which stemmed from a Rojo slip, seemed to anger Mourinho and the Portuguese introduced Zlatan Ibrahimovic, Memphis Depay and Ashley Young in a triple change in the 63rd minute.
Martial, Rashford and Mata, who started as the front three, were all replaced as Mourinho put a new attack on the field, and it almost paid dividends as Young delivered onto the head of Ibrahimovic in the 65th minute, but the referee blew his whistle for a free kick after the Swede jumped with Eric Botteghin.
Bilal Basacikoglu, who came off the bench in the second period, was next to come close for the home side after working some space outside the Man United box, but his effort was always wide of De Gea's post.
Young again found Ibrahimovic 15 minutes from time as Man United looked to make the breakthrough, but the Swede's header was wide of the post as it remained all square entering the final stages of the clash in Rotterdam.
It was Feyenoord that made the breakthrough in the 79th minute, however, when Nicolai Jorgensen broke down the right before delivering a low cross into Vilhena, who found the back of the net with a smart finish.
Smalling was deployed as a centre-forward for Man United in the latter stages and his presence created a chance for Young in the 84th minute, but the Englishman could only slice his effort wide of Jones's post, before Jones kept out a late Ibrahimovic free kick as the Red Devils suffered defeat.