Comrie slipped a backhander through Niklas Backstrom's pads in the sixth round of the tiebreaker, and Jeff Deslauriers deflected Guillaume Latendresse's shot over the net with his glove to finish off the Wild.
Latendresse scored in regulation for Minnesota.
The Oilers, last in the NHL, snapped a three-game losing streak.
Comrie opened the scoring at 6:50 of the first period, swinging around behind the net and hooking the puck behind Backstrom.
Latendresse tied it at 7:34 of the second. Latendresse went around defenseman Tom Gilbert and swung behind the net before calmly depositing the puck behind Deslauriers.
Minnesota had a power play for the final minute of overtime but couldn't capitalize, sending the game to the shootout.
Notes: The home team won all four games in the season series. ... The Wild have more wins (27) against the Oilers than they have against any other NHL team.