Brunette scored on a one-timer in the first period and added a redirect midway through the third for Minnesota's second power-play goal in 32 chances. Niklas Backstrom stopped 33 shots to end a four-start losing streak and Brent Burns added a goal to help the Wild to their first win in Phoenix since 2008.
Phoenix made it interesting, getting Taylor Pyatt's second goal with a two-man advantage with just under a minute left, but couldn't generate any more good chances in its fourth loss in six games.
Minnesota got the season off to decent start, but went into a funk starting in late November.
The Wild came into the game Thursday with just one win in eight games, though they did rally from deficits to pull out a point against Calgary and Dallas in the past two.
Against Phoenix, the Wild didn't take long to get their offense going, generating some good early chances before Brunette broke through with his sixth goal 7 minutes in. He used a deke to flip a backhander past Ilya Bryzgalov on a 2-on-1 after the Coyotes got caught bunched up near mid-ice.
Pyatt tied it late in the period, one-timing a nice feed from behind the goal by Oliver Ekman-Larsson, seconds after the defenseman jumped out of the penalty box.
Minnesota withstood an offensive flurry by Phoenix, thanks to Backstrom and a puck that ran up his shoulder off the crossbar, and Burns put the Wild back up with just over a minute left on a one-timer from just outside the crease on a through-traffic cross from Miettinen.
The Wild spent a good chunk of the second period in their own end and came out of it still with a one-goal lead after Backstrom turned away 10 shots and Coyotes center Shane Doan missed the goal on two good chances. Hanzal also whiffed on a shot from close range later in the third period before getting the redirect late.
NOTES: Wild D Clayton Stoner (upper body) returned to practice this week, but wasn't in the lineup, missing his sixth straight game. ... Brunette's first goal ended Bryzgalov's shutout streak at 101:29. ... Phoenix D Keith Yandle notched his 100th career point on Pyatt's second goal.
(This version CORRECTS Wild 3, Coyotes 2. Corrects spelling of Taylor in third paragraph.)