Kaprizov made it 1-0 off a pass from Mats Zuccarello and banking it off an Arizona skate for a 1-0 lead at 5:47 of the third.
"It's fun for the team and for Kirill to score three goals," Zuccarello said. "It's a big day for him. But most importantly it's a huge win for us at the end of the day."
Kaprizov scored at 8:10 to make it 2-0, and with 49 seconds left for the 4-0 final. The forward played his 25th NHL game.
The 23-year-old, who uses a translator to speak to the media, was unavailable after the game.
"He's hanging out with Mats a lot here and taking some tips from a great professional," Wild captain Jared Spurgeon said. "He works hard every day and he loves to come to the rink. It's fun to see a guy like that get rewarded. He's having a fun time out there."
Jonas Brodin scored into an empty net at 19:01 to make it 3-0.
"We got off to a great start," Hill said. "We had some early pressure, got some shots on net, had some good chances. I think we hit a couple of posts tonight. Just unfortunate bounces."
NOTES: Kaprizov leads the Wild and NHL rookies with 10 goals and 23 points. He is the second Wild rookie to score a hat trick (Justin Fontaine, Jan. 9, 2014) and the third Wild to score one in one period (Zach Parise, March 24, 2016; Jason Zucker, Nov. 9, 2017). ... Minnesota forward Marcus Foligno left the game at 11:22 of the first period after appearing to take a slap shot to his right ankle. He had scored eight points (three goals, five assists) in his previous six games. Evason had no update on his status. ... Zuccarello has scored 17 points (four goals, 13 assists) in his 14 games this season. ... The Wild have scored at least three goals in 17 of 25 games this season and are 15-1-1 in those games.