Hackett replaced starter Josh Harding 71 seconds into the game and the Wild won their fifth straight on the road. Minnesota won for the 12th time after giving up the game's first goal.
Joe Pavelski scored for the Sharks, who have lost four of their last five games.
Harding left with an apparent neck injury 11 seconds after allowing goal in the first minute of play. He was hit in the face by teammate Nick Schultz, who was trying to block Pavelski from the net.
Bouchard's power-play goal, with 12 seconds left in the first period, was set up by Koivu's perfectly placed pass from off the right board. Bouchard wristed it inside the near post for the go-ahead score.
Koivu tied the game with 2:19 left in the first period after Dany Heatley's shot bounced off Niemi's pads to Koivu's stick.
Wild forward Casey Wellman, who grew up an hour from San Jose, recorded a point while playing in front of family and friends for the first time. His father, Brad Wellman, was a major-league infielder and played five years with the San Francisco Giants.
Wellman tipped the puck forward during a line change that led to Koivu's goal for his fourth point in three games.
The Sharks took the lead in the first minute of play off a face-off. Pavelski won the drop, with Dan Boyle slapping a spot off the pads of Harding. Pavelski was all alone and found a hole along the ice.
NOTES: The Sharks had 16 shots before the Wild took their first shot of the game. ... Boyle snapped a nine-game scoreless streak with his first period assist. ... Pavelski has three goals in his last 12 games after scoring nine in his first 12. ... Sharks D Doug Murray missed the game with a hand injury. ... Wild F Devin Setoguchi (lower body) left the game early in the first period and did not return. ... Hackett's uncle, Jeff Hackett, was the Sharks' original goalie. ... Bouchard scored his first power-play goal in three years. ... The Wild improved to 10-0-3 when leading after two periods. ... The Sharks fell to 1-8 when trailing after two periods. ... The Wild are one of five NHL teams not to be shut out this season