Gustavsson made five saves in overtime before denying Simon Holmstrom, Bo Horvat and Kyle Palmieri in the shootout.
Frederick Gaudreau scored the only goal in the third round for Minnesota, which is 17-7-6 in one-goal games this season.
"It's just how the games are though. Like we win with one goal, I don't know how many games this year it's only been [by] one goal," Gustavsson said. "So yeah, I've trusted the process there. We've been lucky enough to get a few more points more than we lost points, so it's great."
Ryan Reaves scored his first goal of the season for the Wild (34-21-6), who have won their past two games and are 6-0-1 during the point streak.
"Our defensive zone has been pretty locked down lately," Reaves said. "We're staying out of the box, which is good for rolling lines and keeping everybody in it. I think we're starting to get comfortable with these low-scoring games where we're just locking down the [defensive] zone and putting a couple in the net."
Ilya Sorokin made 30 saves, and Josh Bailey scored for the Islanders (31-25-8), who have points in four of five (3-1-1). They are 0-5 in the shootout this season with one goal in 14 attempts.
"Guys got to find a way to score," New York coach Lane Lambert said. "It's them and the goalie. They've got to find a way to score, that's all there is to it. That's why. We're not scoring goals in the shootout. We're not winning the shootouts. They have to find a way."