Marco Larsen brought an early save from Keiren Westwood but it took Johnson just six minutes to give the Black Cats the lead.
The former Manchester City man gathered a pass from Jozy Altidore and fired past goalkeeper Jonas Lossl from the edge of the box.
Frank Kristensen almost had the hosts level midway through the first half but Westwood did well to save his close-range header.
Emanuele Giaccherini brought a save from Lossl as half time approached but Pione Sisto almost had the hosts ahead just after the break as Westwood was forced into a low save.
A host of substitutions followed with Midtjylland changing their entire XI but it was Stephane Sessegnon who came closest to scoring again as he hit the bar with seven minutes remaining.