Brunner set up Johan Franzen's tally 2 1/2 minutes into the first period and then found the net himself late in the frame, putting the Red Wings on the path to a key victory.
Franzen added an empty-netter while Valtteri Filppula also scored and Henrik Zetterberg supplied three assists in the win.
Howard turned aside all 34 shots he saw en route to his fourth shutout of the year for Detroit, which moved within one point of a Western Conference with three games to play.
Mike Smith made 19 saves for the Coyotes.
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