The Red Devils were without a number of injured players and lost David de Gea in the warm-up but took the lead thanks to Memphis Depay’s opener in the 37th minute.
However, the home side drew level through Pione Sisto in the 44th minute, before Paul Onuachu struck the winner 13 minutes from time.
Speaking on BT Sport after the defeat, former United midfielder Hargreaves said: “The result is a poor one. The only positive is that they got an away goal.
“Defensively they were all over the place and Onuachu came on and had a big impact on the game.
“Midtjylland wanted to win this game and I think they saw that Manchester United were there for the taking.
“That’s the biggest surprise – Midtjylland deserved to win this game, no question.
“They were positive and created chances from open play and had double the shots on target than Manchester United in this game and I think, sadly to say, right now this [the defeat] is not a surprise.”
The defeat followed United’s 2-1 loss to Sunderland in the Premier League at the weekend and will only heap more pressure on under-fire manager Louis van Gaal ahead of Monday’s FA Cup clash with Shrewsbury Town.
The Red Devils are currently fifth in the Premier League table and six points adrift of the top four.
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