The Sharks were 0-3 in the shootout entering Sunday.
"(It feels) pretty [darn] good," Sharks coach David Quinn said of the win. "Hasn't happened yet, obviously. Not only just the shootout, but we stuck with it. It was a tough first period.
"We could have easily got really down and packed it in after we went down 2-0, and we didn't. We kept coming. … Just really happy and proud of our guys. We really competed, and that was a gutsy win."
Frederick Gaudreau and Connor Dewar scored for Minnesota (7-6-2), which had won two in a row. Filip Gustavsson made 35 saves.
"I hate to say you get what you deserve or whatever, but we got what we asked for tonight," Wild coach Dean Evason said. "I don't know any other way to describe it. It was a dumb loss. We played so east, west, so cute. Not straight forward. All of the cliches. We didn't deserve to win that hockey game tonight."