Mikael Granlund scored for the Wild (2-2-2), who won in regulation for the first time this season.
"I've learned that winning is very difficult in this league, so every single one is very enjoyable," said Dubnyk, who is 2-1-2 with a 2.32 goals-against average and .940 save percentage in five games. "It stings that much more losing in overtime or tying it late and losing in a shootout. Those hurt. To be able to hang on feels really good."
Brendan Perlini scored on the power play to give the Coyotes (1-4-0) a 1-0 lead at 13:12 of the first period.
Arizona was shut out in three of its first four games and has been outscored 11-3 this season. It has not scored an even-strength goal.
"Once again, we scored one goal. We got to score some goals," Coyotes coach Rick Tocchet said. "We had some opportunities there, a couple point-blankers, and we didn't put them in. You saw the game. There's some empty nets there. I don't know what else to say. … I mean, the play is there, put it in, you put it in. There's no secret."
WATCH: [Coyotes vs. Wild highlights]
Granlund tied the game 1-1 with a long-range wrist shot at 3:40 of the second period. He has two goals and two assists in a three-game point streak.
Former Wild goalie Darcy Kuemper made 26 saves for the Coyotes. It was the first time he faced Minnesota, which drafted him in the sixth round (No. 160) of the 2009 NHL Draft.
"I spent five years here, so it was my first time back at all, so to get the start, I was excited, anxious to get going," Kuemper said. "I was really hoping for a win, just couldn't make it happen today."
The Wild played without forward Mikko Koivu, whose wife went into labor prior to the game. Minnesota forward Matt Hendricks left the game with 56 seconds left in the first period with an apparent leg injury.
