Memphis Depay sent the away side into a 37th-minute lead in Denmark, but goals from Pione Sisto and Paul Onuachu ensured that the Danish side would take an advantage into the second leg at Old Trafford next week.
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Man United boss Louis van Gaal made four changes to his XI and there was a full debut at right-back for Donald Love, while Paddy McNair, Ander Herrera and Memphis were all handed starting roles for the English club.
It was the home side that put the pressure on in the opening 10 minutes, with Sisto demonstrating his excellent ability in a wide position, while Kian Hansen's bullet of a throw-in caused some problems inside the Man United box.
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Romero was then called into action down the other end as the Argentine kept out a point-blank header from Hansen, which looked certain to be sending the hosts ahead.
Chris Smalling was next to come close for Man United in the 21st minute after rising to meet a Daley Blind corner, but the centre-back's header was cleared en route to goal, before Anthony Martial was denied a possible penalty after going down inside the Midtjylland box under a challenge from Rilwan Hassan.
Man United took control of the possession as the first period entered its latter stages and might have been ahead when Blind delivered a dangerous cross in to Martial, but the striker could not make clear contract with his head.
Midtjylland had a wonderful chance to score in the 36th minute when Vaclav Kadlec went through one-on-one with Romero after a quick counter, but the away side's goalkeeper made a fine stop and less than one minute later, Man United were ahead when Depay finished from close range after Jesse Lingard had delivered from the right.
The home side were on level terms in the 44th minute, however, when Sisto intercepted a loose pass from Michael Carrick before carrying the ball to the edge of the Man United box and angling his effort into the bottom corner, albeit with the help of a deflection from Smalling.
Man United had the first opportunity of the second period when Martial set up Lingard outside the Midtjylland box, but the Englishman fired into the stands with a poor attempt.
Mata was next to come close for the visitors in the 53rd minute after a cross from Depay, but the Spaniard headed wide of the post from inside the box, before Lingard rattled the crossbar two minutes later.
Midtjylland must have thought that they had a second just past the hour when Onuachu powered a header towards the top corner, but Romero came from nowhere to save.
Sisto continued to threaten the Man United backline and again came close 20 minutes from time when he curled one towards the top corner, but it just missed the far post.
The home side took the lead for the first time in the match in the 77th minute, however, when lively forward Onuachu turned on the edge of the Man United box before finding the bottom corner to send the supporters into dreamland.
It could have been 3-1 moments later when Sisto beat Blind and looked for the top corner, but his effort was just too high.
The expected response from Man United just did not arrive, with the visitors struggling to match the energy on display from the Danish outfit, which was superior throughout the 90 minutes.
Four additional minutes were added and Carrick came close with a near-post header for Man United, but the home side held on to hand themselves a strong chance of making the round of 16.