Brock Boeser scored, and Thatcher Demko made 34 saves for the Canucks (24-32-5), who have lost six of nine.
"We didn't have an answer for No. 97," Vancouver coach Rick Tocchet said of Kaprizov. "Man, he's a [heck] of a player; he was a difference maker. Guys are trying. He's a world-class guy. I think we could have played through him a little bit more. You can't [stop] him, obviously, he's shifty. But I think we could have doubled up a little bit more in the [defensive] zone."
Demko was making his second straight start after missing almost three months with a groin injury and said he felt better than during a 5-4 overtime win at the Dallas Stars in his return on Monday.
"I tried to take a couple more steps tonight, which I thought I did," Demko said. "I still have some steps to take, but I felt good. Obviously, we want to get the win and we knew it was going to be a tight, low-scoring game and it didn't go our way, but I thought we did some good things."
Kaprizov put Minnesota ahead 1-0 just 44 seconds into the game with a backdoor tap-in after he was left unchecked to Demko's right and took a pass from Mats Zuccarello at the top of the right face-off circle.
"That guy has been unbelievable for us," Fleury said.
Boeser tied it 1-1 on the power play at 14:03 with a one-timer from just inside the left dot after a cross-ice pass from Vitali Kravtsov.
It was Kravtsov's first point with Vancouver since being acquired from the New York Rangers on Saturday.
"It was a great play by him," Boeser said of Kravtsov. "I think he saw the lane, made the pass and luckily I put it in the net. We weren't great in the first. In the second period, we responded well, and really just couldn't get anything going in the third.
"There are definitely some parts of our game that are pretty sloppy and we gave them some odd-man rushes, so we definitely need to clean up that aspect of it."
Kaprizov put Minnesota ahead 2-1 at 3:40 of the second period on a breakaway, converting his own rebound after Demko stopped the first shot.
"He's a superstar in this league for a reason," said Wild forward Ryan Reaves, who had the secondary assist on the goal. "Obviously there's games where he's the only guy scoring, so he's a very elite player."