It took Niklas Backstrom to cool them off.
"All over the ice, we played a smart game," Backstrom said. "We're playing some good hockey, and we have to keep building like this."
The Sharks held an 18-4 shots advantage midway through the second period, but found themselves trailing after Brunette scored a 5-on-3 goal with 7 minutes left in the period.
"Sometimes the shot clock doesn't really tell the whole story," Wild defenseman John Madden said. "They had some power plays and have some guys that can find lanes and get pucks to the net. They're a shoot and crash team."
Harassed by Wild defensemen Brent Burns and Nick Schultz, and denied by Backstrom all night, Thornton, Heatley and Marleau were shut down. Backstrom recorded his 20th career shutout and first since March 3, 2010, at Calgary.
"We had a few good looks, but there wasn't an abundance of second and third chances. He did a good job," Sharks forward Joe Pavelski said.
The Wild killed five power plays, highlighted by a key stop with 7 minutes remaining in the third. The Sharks entered as the top power-play team in the league at 31.6 percent.
Overall, Sharks coach Todd McLellan seemed pleased with his team's play, except for the power play.
"That's the one area where I was disappointed," he said. "It looked like we had a lot of hoping instead of knowing. We were just a bit off."
The win snapped a five-game losing streak for the Wild against the Sharks and snapped Backstrom out of a funk against San Jose.
Before the shutout, Backstrom was 2-7-3 with a 3.19 goals-against average vs. the Sharks. The Fin made several key saves on Sharks power plays, including denying Joe Pavelski in the third on a 5-on-4 advantage.
After starting slow, the Sharks had won four of the previous five and scored 20 goals in the four wins.
"In an 82-game season, you're going to have five or seven games that you throw away because you're just that bad," McLellan said. "Tonight wasn't one of those nights. I thought we played well, but Backstrom was better."
The Wild had just six goals in their last four games, but managed to do enough to earn another important win against a quality opponent. The Wild also had a big victory over the Washington Capitals last Thursday.
Brunette scored by taking a pass in the crease from Martin Havlat and beating goalie Antero Niittymaki. Brunette had fanned with an open net seconds earlier, and looked relieved after making the most of his second chance.
The power-play goal was the Wild's second in their last 20 chances.
Niittymaki made his third straight start and stopped 15 shots.
Wild forward Pierre-Marc Bouchard skated at full speed during practice Monday, but did not set a date for his return from a concussion. Bouchard has been out since Oct 6, 2009. ... Brunette's goal was the first 5-on-3 goal allowed by the Sharks this season. ... Wild D Cam Barker returned after missing the last four games with a groin injury. ... Sharks coach Todd McLellan moved C Torrey Mitchell from the second line to the third and dropped Devin Setoguchi to the third. ... Several Wild players have grown mustaches to support Movember, an annual charity event in November that encourages men to grow mustaches to raise awareness of cancers affecting men. Cal Clutterbuck's mustache even has its own Twitter page.