(JIM MONE ~ Associated Press)
Koivu tied the game with 1:38 to play with a power-play goal. He then scored in the shootout.
Minnesota's Josh Harding turned away Kevin Shattenkirk in the tiebreaker, and the Northwest Division-leading Wild won for the ninth time in 11 games.
The Wild's 27 points are the team's most through 20 games in its 11 NHL seasons.
T.J. Oshie and David Backes scored for St. Louis, which has lost four consecutive road games.
The Blues also lost to the Wild on Nov. 5, and they fired coach Davis Payne the next day. St. Louis is 4-0-2 under new coach Ken Hitchcock.
Oshie make it 2-1 with 13:53 to play in regulation, but the Wild continued their late-game heroics.
St. Louis took a 1-0 lead halfway through the first period on Backes' seventh goal.
Oshie fired a slap shot and Backes poked in the rebound seconds after Wild goalie Josh Harding denied Jason Arnott from close range.
Koivu tied it at 2:37 of the second with a short-handed goal, his second of the season and his first at home since March 20.