Kyle Brodziak scored twice in the third period and Koivu in the second period for Minnesota, which for the second straight game overcame a 3-1 third-period deficit to win.
The Wild rallied to win 4-3 Tuesday at Boston, one night after allowing five third-period goals in a 5-4 loss against the New York Rangers.
"It goes to show that you've got to play a full game to win, and we've been able to bounce back the last couple games. We didn't get off to the best start, but we hung with it and found a way," Pominville said.
Joe Thornton, Tommy Wingels and Mirco Mueller scored for the Sharks, who earned five of a possible six points on a road trip that included wins in Anaheim and Colorado. Antti Niemi made a career-high 43 saves.
After Logan Couture scored in the shootout for San Jose, Koivu scored on a backhand to tie it after two rounds. Pominville scored with a wrist shot in the third round before Wild goaltender Darcy Kuemper stopped Patrick Marleau.
Brent Burns nearly won it for San Jose with 1:10 left in overtime as he charged down the left side and drove to the net where he bumped into Kuemper as the puck trickled toward the net.
However, the net became dislodged before the puck crossed the line.
"I just kind of got pushed into the net so I really didn't see it. I knew it was in. They said I pushed the goalie, so I don't know. Can't say I agree," Burns said.
Minnesota was 0 for 1 on the power play, to extend its season-opening drought to 27 opportunities with the extra man. The Wild are the only team without a power-play goal.
With the Sharks leading 2-1 after two periods, Joe Pavelski and Thornton executed a 2-on-1 break early in the third period that ended with Thornton putting a wrist shot from the left circle over the glove of Kuemper and just inside the far post.
But Minnesota answered at 5:13 when Brodziak tipped a shot from Thomas Vanek behind Niemi to make it 3-2.
The same pair hooked up midway through the period to tie the game with Vanek sending a pass to Brodziak, a healthy scratch in five of the Wild's first eight games, charging down the slot for a tip behind Niemi.
Vanek, who signed a three-year, $19.5 million contract on July 1, has six assists in his past five games and seven for the season. He has yet to score a goal.
"He's an incredible passer," Brodziak said. "He's got great vision and he sees the ice very well. I pretty much knew both of those ones were going to be on my tape."
Mueller scored his first career goal late in the first period to give San Jose a 1-0 lead.
His centering pass from near the left corner deflected off the stick of Zach Parise in the crease and between the pads of Kuemper. It marked the first time Minnesota has trailed in four home games.
Wingels made it 2-0 with a power-play goal at 4:20 of the second before Koivu cut the deficit in half when he tipped home a pass from Jason Zucker midway through the period.
"We showed signs of fatigue a little bit," Sharks coach Todd McLellan said. "Then when the fatigue stepped in, we were not mentally clear on some of the decisions we made and we fed them."
NOTES: Wild D Jared Spurgeon left the game in the third period after a check into the boards. Coach Mike Yeo did not have an update on his condition. ... San Jose has outscored its opponents 14-1 in the first period. ... Wild LW Matt Cooke missed the game with a lingering lower-body injury that Yeo said is not serious. ... Minnesota recalled LW Stephane Veilleux from AHL Iowa and reassigned D Stu Bickel to Iowa. ... Sharks C Barclay Goodrow made his NHL debut. He was activated Tuesday after injuring his left hand blocking a shot in the final exhibition game. ... Wild C Erik Haula missed his second straight game after taking a shoulder to the head from the Rangers' John Moore on Monday. Moore was suspended five games for the hit. Yeo said Haula might return Saturday.