Memphis Depay opened the scoring in the first half before Pione Sisto hit back only six minutes later to leave the scores level at half time.
But thirteen minutes from time FC Midtjylland deservedly took the lead as substitute Paul Onuachu downed Louis van Gaal’s side.
That means the Danish side take a slender lead to Old Trafford ahead of the second leg next week.
Manchester United suffered a massive blow during the warm up when goalkeeper David De Gea turned his ankle while making a save, meaning Sergio Romero took his place between the sticks.
Depay had the first chance of the game when Jesse Lingard sent him free but his shot was well blocked by Kian Hansen.
Hansen nearly put FC Midtjylland ahead at the other end but Romero pulled off a stunning save after Daley Blind lost his man at the corner.
And Romero was in action again when FC Midtjylland cleared for Vaclav Kadlec to run clear through on goal, only to fire the ball straight at the legs of De Gea’s replacement.
Only 56 seconds after that golden chance and Manchester United took the lead as Depay met Lingard’s cross and managed to poke the ball into the net, despite slipping after his first touch got away from him.
United’s lead lasted a mere six minutes though, Sisto intercepted Michael Carrick’s slack pass before diverting the ball into the net via the shins of Chris Smalling.
After the break Anthony Martial went on a mazy run before laying off to Lingard who, despite being in acres of space, sliced the ball over the bar.
Lingard had another golden chance to score on 55 minutes but his volley from Ander Herrera’s cross smashed against the bar when he really should have scored.
Onuachu – who stands at 6ft 7in – came on after the hour mark and within seconds had forced Romero into another fantastic save from his towering header.
But on 77 minutes Onuachu did beat Romero as Mata pulled out of a tackle on the edge of the area and the Nigerian striker fired into the back of the net.
That goal still could not spark a lacklustre Manchester United side into action with FC Midtjylland looking the more dangerous of the two as the Danish side secured the advantage ahead of the second leg.