Ross Johnston scored his first of the season on a breakaway at 4:03 to cut it to 3-2.
Kaprizov redirected Matt Dumba's shot to make it 4-2 at 11:27 of the third period.
Oliver Wahlstrom dove to knock in a loose puck during an Islanders power play at 16:00 for the 4-3 final.
"They obviously pushed very hard, but I think we had a great battle tonight," Kahkonen said. "It's not always pretty and you've got to just find a way to get the job done, and that's what we did."
New York outshot Minnesota 14-6 in the second period and 18-5 in the third to finish with a season-high 43 shots on goal.
"I thought we stayed in the fight," Trotz said. "We had a forecheck, we stayed and created a lot of different things off the forecheck. … Defensively, they really didn't get much. They got a couple looks. I thought we did a really good job containing, especially their top line (Kaprizov, Ryan Hartman, Mats Zuccarello), they end up getting the winning goal, but for the most part we had contained them all night. They didn't get much. … We had tons of chances and we just, we deserved a point tonight. We really did."
Zuccarello had his 10-game point streak end. He scored 18 points (five goals, 13 assists) in his run and multiple points in a Wild record six consecutive games (13 points: four goals, nine assists).
NOTES: Kaprizov has a personal 11-game point streak (eight goals, 12 assists). … Wild forward Marcus Foligno did not play because of an upper-body injury sustained during a 3-2 win at the New York Rangers on Friday. He is day to day. … Nelson has a three-game point streak. He has three goals and three assists in his past five games. … Islanders center Casey Cizikas was a late scratch due to a non-COVID-19 illness. … It was Johnston's first goal since Dec. 27, 2019.