Kaprizov tied it 2-2 seven seconds into a power play, scoring from the slot at 10:53.
Frederick Gaudreau gave Minnesota a 3-2 lead at 13:59 with a backhand at the left post.
Sam Steel increased the lead to 4-2 at 3:13 of the third period, and Zuccarello tipped a Matt Dumba shot from the blue line at 11:48 to make it 5-2.
"All four lines can score for us," Steel said. "You see it almost every night. Everybody's contributing, so that's huge for the team. We're just hoping to sustain that."
Klim Kostin scored his first goal of the season with four seconds remaining for the 5-3 final.
"I think when we got the lead, what we did good today, we kept pushing," Eriksson Ek said. "We didn't sit back, and I think we kept playing and just really held on to the puck and didn't give too much. It was important for us to keep going and didn't sit back."
NOTES: Kaprizov got his 100th and 101st assists in his 158th NHL game, the fewest number of games to reach 100 assists among players to make their debut with the Wild (previously Mikko Koivu, 230 games). … Fleury (24,771 saves) passed Tony Esposito (24,761) for the fourth most in NHL history behind Martin Brodeur (28,928), Roberto Luongo (28,409) and Patrick Roy (25,800). … Wild forward Matt Boldy had his three-game goal streak end. … Eriksson Ek has nine points (three goals, six assists) in his past six games. … McDavid's assist was his 482nd, tying Bryan Trottier for fifth most by a player age 25 or younger. … The Oilers, who average 32.7 shots per game, were held to a season-low 21. … The Wild are 7-0-0 against the Oilers dating to Oct. 22, 2019.