Brodziak put the Wild ahead to stay with a goal 2 minutes into the game on a diving swipe that found its way past Coyotes goalie Mike Smith.
He made it 3-0 midway through the middle frame on a wrap-around shot during another man-advantage.
Niklas Backstrom, making his first appearance since Dec. 2 due to a strained groin, stopped 35-of-36 shots to earn the victory.
Lauri Korpikoski spoiled his shutout bid with a third-period goal and Smith finished with 18 saves for the Coyotes.
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