Minnesota (31-13-3) got the win it needed to secure its place as one of four qualifying teams from the eight-team Honda West Division. It will be their eighth playoff appearance in the past nine seasons.
"[It's] nice to not have to wait until the last week," Wild captain Jared Spurgeon said. "Not that the games don't mean anything for the rest of the season, but just to know that you're in a position and that you're in the playoffs.
"But that's just the first step of what we want to do. It was nice to do it tonight, for sure."
Kirill Kaprizov extended his goal streak to five games, and Kaapo Kahkonen made 27 saves for the Wild, who have won seven in a row.
Minnesota moved within one point of the second-place Colorado Avalanche having played two more games. Colorado and the West-leading Vegas Golden Knights are the three teams to have clinched a berth this season.
"There's not one of us who doesn't think we can't be No. 1 in our division," Wild forward Marcus Foligno said. "It's still in reach (five points behind Vegas). … There's great teams in our division, but we've had success against them (4-1-1 vs. Vegas, 3-5 vs. Colorado)."
Joachim Blichfeld scored his first NHL goal for San Jose (18-24-5), which has lost eight straight and is sixth in the West, six points behind the fourth-place Arizona Coyotes, who defeated the Los Angeles Kings 4-0 on Saturday.
Martin Jones allowed three goals on eight shots in the first period and was replaced by Josef Korenar, who made 15 saves.
"The start was a disaster," Sharks coach Bob Boughner said. "Right from the first shift, the first goal. We're fighting for our lives, and to be down 1-0 on the first shot of the game was a horrible goal. I think it took some wind out of our sails, but that's an excuse as well. You have to be able to rebound, you've got 59 more minutes to play, and I don't think we did a good job at coming back from that adversity.
"You need saves at the right time in the game and we didn't get them early, and now you're chasing the game. You'd love to come out of that tied or leading after one and build your game and keep getting better, but we didn't get the chance to do that."