The visitors were quickest off the starting blocks as, just four minutes in, Jose Fonte fired goalwards from a corner, but the ball was headed off the line.
Southampton continued to push forward and create chances, but they were caught out in the 23rd minute as the ball was given away to Morten Rasmussen, who found space in the box before finishing low past Maarten Stekelenburg.
Midtyjlland then almost doubled their lead on the stroke of half time when Pione Sisto curled the ball over from the left of goal, and carried on with intent after the restart as Rasmussen headed wide from a free kick.
The hosts, however, held firm at the back, repelling each and every Southampton cross and shot that had the slightest hint of causing problems.
By the latter stages of the game, the Saints were reduced to ineffective long-range balls into the final third, and Midtjylland succeeded in seeing the game out to reach the Europa League group stages at the South Coast club's expense.