Mike Zigomanis tied it midway through the third period and Perreault added his third goal in two games for the Coyotes, who won consecutive games for the first time this season. After Joseph stopped Radim Vrbata with a pad save, Perreault faked goalie Brian Boucher to the ice and fired the puck into the top of the net. Jeff Hamilton evened the shootout, also shooting high over Joseph.
But Nagy beat Boucher with a sharp wrist shot low to the stick side, and Joseph made a pad save on Mikael Holmqvist's forehand attempt to secure the win.
Karl Stewart and Martin Lapointe scored 92 seconds apart midway through the first period for the Blackhawks, who are 2-0-1 in their last four games.
Phoenix tied the game with 11:19 remaining when Georges Laraque worked the puck free in the right faceoff circle and fed a pass to Zigomanis in the low slot.