United were without a number of injured players, including Wayne Rooney, and lost David de Gea in the warm-up.
They took the lead through Memphis Depay's shot but Sergio Romero, who stepped in for De Gea, was unable to prevent Pione Sisto equalising.
United's Jesse Lingard hit the bar before Paul Onuachu struck the winner.
After losing to Sunderland 2-1 on Saturday, the Red Devils are six points shy of the top four in the Premier League, leaving qualification for next season's Champions League very much in the balance.
Winning the Europa League could represent their best chance of reclaiming a place in Europe's elite club competition, but if they are to do that they will have to be much better than they were against opponents who had not played for 71 days because of the Danish winter break.
That includes in the second leg of this tie, which is at Old Trafford in a week's time.
Midtjylland were only formed in 1999, the year United won their famous Treble, but what they lack in history they made up for with application and ability.
De Gea injury begins bad night for United
Manchester United can form an entire starting XI from their injured players
With 13 first-team players out and facing a side whose analytical tactical approach makes them a serious threat from set-pieces, United suffered a major setback when goalkeeper De Gea injured himself in the warm-up.
That meant a first United start in four months for Romero, but the Argentine proved an able replacement with three good saves.
One was a block to deny Vaclav Kadlec, who ran through unchallenged, just 56 seconds before United broke the deadlock through Depay's close-range finish from Lingard's cross.
Romero also produced a superb one-handed diving stop to keep out a powerful goalbound header from Onuachu in the second half.
Between those two saves, he could not stop Sisto pulling the hosts level with a shot that found the net via the shin of Chris Smalling.
Romero was equally powerless to keep out Onuachu's winner - a low drive after the substitute beat Juan Mata on the edge of the box.
Life without Rooney
Memphis Depay (left) scored his sixth Manchester United goal against FC Midtjylland
A knee injury means United face a six-week insight into life without captain Rooney - an absence that could become permanent if reports of a move to China prove correct.
United have refused to comment on speculation about the striker's future, but his absence allowed Depay to make his first start since Boxing Day and gave Anthony Martial the chance to start up front.
Depay, the top scorer in the Dutch league last season prior to his £31m move to Old Trafford, had scored once in his previous 20 United appearances.
And while his display in Denmark was not a notable improvement on past performances in a United shirt he was able to add a goal to his modest tally of goals for the season, which now stands at six.
But the Dutchman, like the rest of the United team, offered little in the second half as the home side came back to win.
Martial, who has operated primarily on the left since joining the club for £36m in September, was largely anonymous in his central role.
Lingard struck the bar with a volley while the score was 1-1, but United rarely threatened as the clock ticked down.
What next?
United travel to Shrewsbury for an FA Cup fifth-round tie next Monday, followed three days later by the return leg against Midtjylland at Old Trafford.
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