On a freezing cold night, the local spectators were not treated to much of a spectacle in the first 45 minutes.
The first half of the semi-final was a very hard-fought encounter with chances few and afar and plenty of stoppages, which constantly disrupted any kind of rhythm in the game.
Shortly before the break, however, the visitors from Central Jutland managed to get the ball in the net after a corner kick, but the goal was ruled out after a free kick on AGF winger Rasmus Carstensen.
The home side enjoyed most of the play during the second half but struggled to create clear-cut chances against a stubborn and well-organized Midtjylland defense.
Instead, the visitors took the lead around the hour mark when AGF failed to clear a cross from Aral Simsir, allowing Han-Beom Lee to head home from close range with AGF goalkeeper Jesper Hansen helpless.
FC Midtjylland followingly dropped deep content just to defend the lead, and despite many promising situations, AGF couldn't find an equalizer.
In the other cup semi-final, FC Copenhagen won the first leg against Viborg on Wednesday with 2-1.
The final will be held in Parken Stadium in Copenhagen on 14 May, Ascension Day.