Marcus Foligno scored twice for Minnesota, which got points from 11 players and won for the sixth time - all by one goal. Caleb Addison and Nico Sturm also scored. Cam Talbot had 24 saves for the Wild, winners of nine straight against the Senators.
Ten players had points for Ottawa. Drake Batherson had a goal and assist. Nick Paul, Josh Norris and Chris Tierney added goals. Michael Del Zotto had two assists. Filip Gustavsson overcame a shaky start and made 38 saves, including 20 in the third period.
On the winner, Kevin Fiala gathered the puck in his own zone, fought off Tim Stutzle into the Senators' end and fed a pass across to Kaprizov. Last season's Calder Trophy winner who signed a five-year, $45 million contract before training camp, went down to one knee and scored from near the right dot.
''I think he said to (assistant coaches Brett McLean and Bob Woods) after the game, it feels like his first goal ever,'' Wild coach Dean Evason said. ''He's pretty excited obviously. It's one thing for him to be excited but to see his teammates that's exciting for us.''
Kaprizov scored 27 goals in 55 games last season. He tied a season high with six shots on goal.
''Nobody on the team was worried about him. We knew it was coming,'' Sturm said.
Trailing 3-1, Ottawa scored three times in less than six minutes of the second period to take a short-lived lead.
Batherson buried a shot from the right circle - a play Minnesota unsuccessfully challenged for a high stick earlier in the shift - and Norris scored from the edge of the blue paint on the ensuing power play to make it 3-3.
''Down 3-1 you could sit back and feel bad for yourselves. . We showed a lot of resiliency and character,'' Del Zotto said.
Tierney, in his 500th game, scored on a rebound when the puck bounced up, deflected off his arm and past Talbot. Initially called no goal, the Senators successfully challenged, and video review showed that Tierney was trying to get out of the way, not direct the puck in.
''I thought I just reacted to the puck coming at me,'' Tierney said.
Foligno converted a pass from Matt Dumba to tie the game 4-4 just over two minutes later.
''They're very highly skilled,'' Foligno said. ''Some great players on that team. We just need to understand that we need to be nasty. We need to be aggressive and desperate. We saw that in the first. We saw that in the third. We're lucky right now that it's two periods of good play that's getting these wins. When we play against some tougher teams that might not be the case.''
Coming off a lackluster 5-1 loss Monday in Chicago, Ottawa gave up the game's first seven shots and was quickly down by a couple goals.
Foligno scored on a rebound and Addison scored his first career goal 27 seconds later when his long shot deflected off Del Zotto and popped over Gustavsson.
Paul redirected a pass from Connor Brown for an Ottawa tally. Sturm scored on a rebound for a 3-1 Wild lead.
NOTES: Wild D Alex Goligoski (upper-body injury) took part in the morning skate but did not play. . Ottawa LW Austin Watson made his season debut after injuring an ankle Oct. 7. He fought Brandon Duhaime in the first. ... Del Zotto played for the first time since Oct. 21. ... This was the first of 32 Hockey Fights Cancer nights scheduled around the league this month.
Senators: Home against Vegas on Thursday.
Wild: At Pittsburgh on Saturday.
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