The Blackhawks forced overtime when Brent Seabrook put in a rebound with only 15 seconds left in regulation, but the Coyotes swarmed the Chicago net throughout the extra time and their persistence finally brought the winning goal.
Adrian Aucoin took a shot from the right point that sailed through traffic. Hanzal was parked in front of the net where he out-muscled Chicago's Marcus Kruger while deflecting the puck past Corey Crawford.
Mike Smith, who allowed only two goals in his final five starts of the regular season and entered the playoffs as a very hot goalie, made 43 saves in the victory.
The score-tying goal late in regulation by Seabrook came on a fluke play in which Smith stopped a shot only to have the puck hit Seabrook's left skate. The puck settled by itself in front of Seabrook and he easily tapped it past Smith to forge a 2-2 tie.
Crawford made 30 saves in defeat.
Phoenix will host the second game of the series Saturday.