Whitney scored on a backhander at 9:43 of the third period to send the game into overtime and then scored the first of two Phoenix goals in the shootout.
Mikkel Boedker beat Cory Schneider on the Coyotes' third try in the shootout to seal the victory, which extended the Phoenix unbeaten streak to 12 games.
Mike Smith stopped 39-of-40 shots for the Coyotes.
Schneider made 34 saves for the Canucks, who lost their second in a row but have only one loss in regulation in their last 18 games.
Alexander Edler gave Vancouver a 1-0 lead in the first period with a power play goal.