Martin Skoula and Brian Rolston also scored for the Wild. Barret Jackman and Lee Stempniak tallied for St. Louis.
The Blues led 1-0 after one period on Jackman's goal, which came at 17:04 in the first period. But Skoula tied it when he beat Blues goalie Manny Legace with a wrist shot 5:34 into the second period.
The Blues were whistled for four minor penalties in a 3:42 span midway through the second period, and Koivu made them pay by scoring two power-play goals.
Koivu gave the Wild the lead at 12:14 when he got control of a loose puck at the right side of the net and wristed a shot into the open goal.
He then made it 3-1 at 14:35 with a one-timer from the high slot.