After battling through a scoreless overtime, Zach Parise gave Minnesota the advantage in the first round of the tiebreaker, but Calgary's Jiri Hudler responded in the top of the second round by putting a shot past Niklas Backstrom.
That set the stage for Koivu, who came in quickly, cut back to a forehand shot and found the net behind Leland Irving.
Backstrom then clinched the win with a pad save on Alex Tanguay as the Wild notched a second straight victory and improved to 1-3-1 on the road.
Kyle Brodziak scored in regulation while Backstrom made 23 stops for Minnesota.
Hudler netted Calgary's only goal and Irving turned aside 23 shots for the Flames.