Scott Perunovich and Jake Neighbours scored, and Jordan Binnington made 24 saves for the Blues (8-11-1), who are 1-5-1 in their past seven games.
"It's frustrating," Neighbours said. "I think we're generating, right. The looks are there. Kind of again the story is we're not finishing on them. That's just the way it's kind of going right now. We just can't seem to finish. I think there's a lot of positive things. We're creating and playing well and we're in games. We've just got to find a way to win."
Hartman's first goal in seven games put the Wild ahead 1-0 at 12:24 of the first period. He scored on a shot from the right circle that trickled through Binnington.
Perunovich tied it 1-1 at 4:48 of the second period, beating Gustavsson over his glove with a wrist shot after he joined a rush and took a pass from Jordan Kyrou.
Kaprizov redirected Jake Middleton's right point shot at 6:07 of the third period to give the Wild a 2-1 lead.
"We had a good, strong forecheck and we created a lot of chances," Minnesota forward Marcus Johansson said. "Every forward line has to be good in the O-zone and D-zone."
Neighbours tied it 2-2 with a power-play goal at 10:37 when he redirected Pavel Buchnevich's pass from the right circle past Gustavsson in the crease.
"Something we looked at in intermission," Neighbours said. "[Assistant coach Steve Ott] spotted it. We kind of drew up a play for it and they weren't really expecting the backdoor a little bit, kind of worried about [Kyrou] in the bumper and all the other options ‘Buchy's’ got over there. Kind of left me alone and made a great play."