After Olli Jokinen's backhander went wide of the net to open the shootout, Whitney skated down the center of the ice and ripped a wrister from the slot that beat Miikka Kiprusoff to the stick side.
Kiprusoff would not get beat again, but the Flames' Alex Tanguay and Jarome Iginla also failed on their shootout attempts as Phoenix improved to 9-0-1 in the month of February.
Whitney, Daymond Langkow and Shane Doan scored in regulation for Phoenix, while Mike Smith recorded his ninth straight win with 21 saves.
Tanguay, Jokinen and Matt Stajan lit the lamp for the Flames, who lost for the third time in four games. Kiprusoff stopped 26-of-29 shots in defeat.