Their much-changed line-up boasted quality but looked cumbersome and unwieldy in Rotterdam, where Feyenoord snatched a 1-0 win through Vilhena's strike.
It was hardly the response Jose Mourinho craved after Saturday's derby defeat at home to Manchester City, with few of the eight changes impressing at the imposing if partially-closed De Kuip.
Anthony Martial's fizzing strike wide was the best United could muster in a lacklustre first half that showed little sign of improvement after the break.
Zlatan Ibrahimovic, Memphis Depay and Ashley Young were thrown on as Mourinho attempted to jump start his side, but their introduction could not prevent United's winless run extending to a sixth European trip.
Instead, they left Holland pointless as the Eredivisie leaders struck a decisive blow as offside calls against Nicolai Jorgensen ignored and his cross was turned home impressively by Vilhena.
It was a disappointing night at the ground where United won the Cup Winners' Cup against Barcelona in 1991, with parts of De Kuip closed this time as Feyenoord attempted to avoid triggering suspended punishments handed down by UEFA for crowd trouble.
A 10-metre high net around the pitch and additional stewards were other measures taken as United started their Europa League campaign with their simplest journey.
Istanbul-based Fenerbahce and Ukraine's Zorya Luhansk lie in wait over the coming months, which is part of the reason why Mourinho chose such a strong side for the trip to Rotterdam.
It was an impressive show of strength but the eight changes from the derby loss struggled, rarely clicking into gear.
David de Gea saved a curling Steven Berghuis effort and easily collected a Vilhena strike during the opening stages, which saw Paul Pogba's free-kick simply saved by Brad Jones.
The former Liverpool goalkeeper had precious little to do in the first half but will have breathed a sigh of relief in the 24th minute when Martial fizzed wide after a Matteo Darmian cross.
It was United's only moment of inspiration in a tepid half that ended with Juan Mata just failing to reach a Marcus Rashford header, having seen Jens Toornstra flash over minutes earlier at the end of a quality Feyenoord move.
Unlike Saturday's Manchester derby the United boss refrained from any changes at the break, which Mourinho emerged from around a minute before his players and staff.
World-record signing Pogba flashed hopelessly wide soon after play resumed, with former Aston Villa midfielder Karim El Ahmadi doing better when coming close with his own long-range effort.
Pogba had another attempt before Mourinho made widespread changes, throwing on Ibrahimovic, Depay and Young for Rashford, Martial and Mata.
Despite the decision to change attack the Dutch side were looking most like scoring, with Jorgensen blocked before Bilal Basacikoglu tried his luck.
Feyenoord fans cheered when Depay missed with a wild attempt, before the noise ratcheted up several notches. Jorgensen appeared offside when put through down the flank, making the most of defensive disarray and cutting the ball back to the edge of the box for Vilhena to turn home.
An incredible din welcomed the goal, with every tackle and missed opportunity cheered in the closing stages. Young failed under pressure and Ibrahimovic stung the palms with a long-range free-kick, with Eric Bailly denied at the near post before the final whistle.
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