Sevilla boosted its quest for a third straight title in Europe's second-tier competition by beating Molde 3-0, while there were also wins for Borussia Dortmund and Villarreal over Porto and Napoli, respectively.
Fiorentina and Tottenham drew 1-1.
Winning the Europa League is regarded as United's best chance of returning to the Champions League next season, but that will be difficult after squandering a lead earned by Memphis Depay. Pione Sisto and substitute Paul Onuachu replied for FC Midtjylland, which was featuring at this stage in Europe for the first time and playing its first competitive game for two months while on a winter break from domestic competition.
The club was formed in 1999 — the year United won the Champions League- Premier League-FA Cup treble.
And it could have been worse for United, with goalkeeper Sergio Romero producing three stunning saves after being drafted in following an injury to De Gea in the warm-up. It adds to United's lengthy injury list that already contained 13 players, including Wayne Rooney, who missed the game after being ruled out this week with a knee injury.
"It's difficult to cope with such a list of injured players," United manager Louis van Gaal said.
"In the second half, I have to say Midtjylland was winning every ball. That is very disappointing — you know in advance that is the key to this game."
Ousted from the Champions League at the group stage and struggling in fifth place in the Premier League, the beleaguered Van Gaal had been looking for some respite from a tough season by winning the Europa League. But Onuachu's 77th-minute winner from the edge of the area, after he beat Juan Mata to a loose ball, means Van Gaal will come under even more scrutiny now.
The second leg is next Thursday at Old Trafford.
"Still we have a chance, I think," Van Gaal said.
Fernando Llorente scored two goals and Kevin Gameiro added a third for Sevilla in a comfortable home win as the Spanish team looks to back up title wins in 2014 and '15.
Dortmund beat FC Porto 2-0 through goals by right back Lukasz Piszczek and Germany forward Marco Reus, and Denis Suarez scored with a free kick off the underside of the crossbar to give Villarreal a 1-0 win at home to Napoli.
Tottenham was pegged back after scoring a penalty through Nacer Chadli, with Federico Bernardeschi grabbing Fiorentina's second-half equalizer.
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