The game-winning play began when Charlie Coyle dug out the puck behind the net and sent it to Nino Niederreiter in the slot, who fired a shot that was blocked by Stars goalie Kari Lehtonen.
But Prosser was there to fire in the rebound for his second goal of the season, handing the Wild their seventh win in nine games.
Erik Haula notched his first NHL goal and Ryan Suter also scored.
Darcy Kuemper earned the victory with 33 saves.
Alex Chiasson and Ryan Garbutt scored for Dallas, which fell nine points behind Minnesota in the race for a Western Conference wild card spot with an eighth loss in nine games.
Lehtonen finished with 16 saves.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys