Mario Ferraro had a goal and an assist, and Mikael Granlund had two assists for the Sharks (19-52-9), who have lost three of their past four (1-2-1). Mackenzie Blackwood made 32 saves.
"At the end of the day, every single one of us [who's] going to be here next year has to take a long, hard look in the mirror and try not to be in the exact same position when we stand here next year," Sharks forward Nico Sturm said.
Matt Boldy gave Minnesota a 1-0 lead at 5:19 of the first period with a one-timer from below the right face-off circle after Kaprizov’s point shot was saved.
Jan Rutta tied it 1-1 at 9:32 with a slap shot from the blue line.
Declan Chisholm put the Wild ahead 2-1 at 16:51 with a wrist shot through traffic from the right circle.
Ohgren made it 3-1 at 14:23 of the second period when he put in a rebound with a backhand at the top of the crease.
"Big moment for me and I’m really happy," Ohgren said. "It feels good, and it will only be better and better."