Kaprizov had his fourth goal of the season and added two assists as Minnesota wrapped up its season-opening, four-game homestand with its first points of the season. Mats Zuccarello added two goals and Sam Steel scored his second of the season.
Marc-Andre Fleury finished with 23 saves for Minnesota.
''To get a win like that, you just believe, or hope and believe that'll push us forward to do what we do,'' Wild coach Dean Evason said. ''We still lost our composure in a lot of areas in tonight's hockey game, which is uncharacteristic, but I guess probably expected when we're squeezing so tight. Hopefully, this allows us to loosen up a bit.''
Bo Horvat scored for the third straight game, Dakota Joshua had a goal and assist and Nils Aman scored his first NHL goal for Vancouver, which let another lead slip away.
Thatcher Demko made 33 saves for the Canucks.
Vancouver had multi-goal leads in each of its first four games, blowing each and coming away with just one point in an overtime loss at Columbus in its previous game. It let another third-period advantage slip away.
''We've either been leading or we've been tied with less than seven minutes to go in every one of the games and we haven't come up with a win, and that's very frustrating,'' Canucks coach Bruce Boudreau said. ''So, we know what we're capable of doing and when we do turn it around, it'll turn around for a long time, and just want to do it quick.''
While the Canucks have had trouble holding leads this season, the Wild had trouble getting in front.
Zuccarello's first goal of the game gave Minnesota its first lead of the new year 4:37 into the first period.
The lead lasted just 3:21 as Horvat scored his fourth goal of the season on a give-and-go, tapping in a centering pass from Tanner Pearson.
Aman, drafted by the Colorado Avalanche in the sixth round in 2020, added his first in his fifth career game. Joshua sent Aman in alone on Fleury with a behind-the-back pass and Aman beat Fleury with a wrist shot.
''Always fun to score,'' Aman said. ''Get a nice pass from Joshua and had some speed, so just trying to get to the net and it went in. But it sucks to lose.''
Steel tied the game before the first intermission. Joshua's goal was the lone tally in the second and held up until Zuccarello scored his second of the game on the power play, deflecting a pass from Kaprizov past Demko.
A team with Stanley Cup aspirations after last season's franchise-record 53 wins and 113 points, the Wild have suffered defensive lapses during their disappointing start. Thursday's game started the same way, but Minnesota recovered.
Fleury, the 37-year-old netminder who re-signed with Minnesota on a two-year deal in the offseason, has allowed 14 goals on 75 shots and was pulled in his previous start. He was at his best from the second period on, allowing one goal on the final 18 shots he faced.
The Wild, who went 31-7-2 at home last season, had allowed 20 goals in the first three games of their opening homestand.
''It's good to see guys laugh a bit, smile,'' Fleury said. ''We have such a good team and chemistry. Everybody kind of got tense, right? Not being ourselves. Nice to be relaxed and smile and having fun playing the game. Working on it. It's good. It's good to see the laugh. We just have to keep building on it.''
Minnesota forward Jordan Greenway made his season debut after recovering from offseason shoulder surgery, but he left midway through the first with an upper-body injury and didn't return.
Evason said Greenway will undergo tests, but the forward won't play Saturday. Evason said it is not the same injury that required surgery.
Vancouver also lost defenseman Riley Stillman to an undisclosed injury. Boudreau didn't have an update on Stillman after the game.
Canucks: Host Buffalo on Saturday in home opener.
Wild: Play first road game at Boston on Saturday.
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