Smith faced only four shots in the third period despite a pair of Minnesota power plays as Phoenix notched a second straight win.
Lauri Korpikoski and Martin Hanzal found the net for the Coyotes.
Zach Parise scored an unassisted tally off a turnover in the second period as Minnesota cut its deficit to one goal, but the Wild could get no closer and fell to 0-3-1 away from home.
Niklas Backstrom made 32 saves in the loss.
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