The Red Devils were facing the Danish champions for the first time without United captain Wayne Rooney and David De Gea, who picked up an injury during the warm-up.
Louis van Gaal’s side made a promising start after Memphis Depay broke the deadlock in the 37th minute at the Arena Herning.
However, the hosts were level before half-time after Pione Sisto beat United’s second-choice goalkeeper Sergio Romero in the 44th minute.
And Midtylland condemned United to back-to-back defeats after Paul Onuacho scored a winner with 13 minutes remaining to put the Danish side in pole position ahead of the return leg at Old Trafford next week.
Former England and Tottenham Hotspur striker Lineker stated his disbelief at United’s poor performance in Denmark.
It's hard to comprehend how poor Manchester United are at present.
— Gary Lineker (@GaryLineker) February 18, 2016
United face Shrewsbury Town in the FA Cup fifth round on Monday night.
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