Jokiharju gave the Sabres a 1-0 lead at 18:02 of the first period on a shot from the blue line that beat Gustavsson through traffic.
Kaprizov one-timed a cross-ice pass by Matt Boldy on the power play to tie it 1-1 at 19:30.
Skinner put Buffalo ahead 2-1 with a breakaway deke to the forehand at 19:02 of the second period.
“Great play by JJ, obviously, to sort of anticipate and poke the puck, and then I was able to get behind their [defensemen] because he kind of poked it by them,” Skinner said.
The Wild had their chances in the second, outshooting the Sabres 14-6. Levi got a piece of Marco Rossi’s backhand in front on a 2-on-1, and Kaprizov later hit the post from the bottom of the right circle.
“We're used to tight games,” Minnesota captain Jared Spurgeon said. “But when you're getting those chances and not putting them home, you have to make sure you're squeaky clean on the other end. Something we’ve got to clean up, but the effort that we had in the offensive zone, I thought we created chances, but we’ve just got to get those looks and make them count.”