Brent Burns and Logan Couture scored for the Sharks (7-8-2). Martin Jones allowed four goals on 20 shots before being replaced in the second period by Devan Dubnyk, who made 12 saves.
"It was one of those games I thought, obviously, our start was good … but every time we had good momentum shifts, I have to give their goalie credit. He made some big saves at the right times," San Jose coach Bob Boughner said. "He made some saves and we just couldn't find the goal we needed at the right time."
Burns scored 45 seconds into the game to give the Sharks a 1-0 lead. He got around Wild defenseman Jonas Brodin before cutting across the slot and scoring with a backhand.
Zuccarello tied it 1-1 at 7:49 with a shot from the right hash marks after waiting for Sharks defenseman Mario Ferraro to slide by, and Kaprizov scored 11 seconds later on a backdoor tap-in to make it 2-1.
"Any momentum builders, you don't want to give up one (goal), you definitely don't want to give up two," Burns said. "Yeah, you don't want to do that."
Ian Cole extended the lead to 3-1 at 3:36 of the second period after receiving a no-look pass from Ryan Hartman, who then found Foligno at 10:48 for a 4-1 lead.
"It's great when a team can score many different ways," Cole said. "We had had a little trouble scoring goals early in the season, but we always knew we had the skill to do it. … Right now, we're starting to figure it out, and we have to keep doing it.
"The 15th game of the year doesn't really mean much. We have to do it the rest of the year and through the playoffs."