Ryan Hartman scored the go-ahead goal early in the third period, and Jared Spurgeon had two assists for the Wild (6-10-1), who were 1-3-1 in their previous five games.
"Obviously, we want to stop falling behind, but we've shown some great character, being able to come back and keep ourselves in the game," Hartman said. "We tend to find our legs and when we do, we're making some great plays."
WATCH: [All Wild vs. Coyotes highlights]
Michael Grabner, Jakob Chychrun and Vinnie Hinostroza scored for the Coyotes (9-6-2), who twice failed to hold a two-goal lead. They are 0-2-1 during their first three-game winless streak of the season and have blown a lead in four consecutive games.
"We've got to buckle down here," Coyotes coach Rick Tocchet said. "We're in control of the game, and we just give them life. It's an unfocused play here or there, and some individuals have to play better."
The Wild are 2-1-0 on a four-game road trip despite falling behind by 2-0 in the first period of all three games.
"You don't want to fall behind and then be chasing the game all the time," said Minnesota goalie Devan Dubnyk, who made 32 saves. "But for the most part we've been playing together. I think we recognize that, and we can still get wins, and in tough buildings."