The Danish champions will take a slender lead to Old Trafford next week after substitute Paul Onuachu fired home a late winner in Thursday's first leg in Denmark.
Injury-hit United, already without 14 players through injury, lost David de Gea in the warm up and were indebted to Argentine stand-in Sergio Romero whom prevented further damage with a string of excellent saves.
United started well and went close early on through Memphis- with the winger's snapshot diverted behind for a corner- and Juan Mata then had an effort saved by Mikkel Andersen from the resultant flag-kick.
Midtjylland were not prepared to simply sit back though and soon threatened themselves through Kian Hansen, whom was inches away from converting a corner of their own, only to be denied by the sprawling Romero.
Memphis forced a good save from Andersen from a free-kick whilst Romero kept out Vaclav Kadlec at the other end after a rapid counter-attack from the hosts.
Just 56 seconds later, United were ahead as Jesse Lingard broke down the right and picked out Memphis with an excellent cross for the Dutchman to scramble in his sixth of the season eight minutes before the break.
The Reds were to go in for the interval only level, though, as Pione Sisto cut out a Michael Carrick pass and fired home past Romero via a deflection off Chris Smalling on the cusp of half-time.
The setback did not seem to unsettle United, whom continued to press and created three good chances early in the second half.
Lingard lashed over from distance and then struck the bar before Juan Mata nodded narrowly wide from a Memphis cross in a spell of pressure from the visitors.
Whilst the woodwork had come to Midtjylland's rescue, it took a piece of brilliance from Romero at the other end to prevent the hosts from regaining the lead.
Onuachu met a Nikolay Bodurov cross at the back post and nodded goalwards only for Romero to dive full-length and tip the header away.
The substitute was not to be denied, though, and slammed home from the edge of the area on 77 minutes to leave United in need of a frantic finish,
Michael Carrick headed over the bar late on, but no late fightback was forthcoming to leave the tie- and our fate- well and truly in the balance.
Overall team performance: 5/10
United Faithful Man of the Match: Sergio Romero