Smith turned aside all 13 shots he saw in the second period in recording his third whitewash of the season and 22nd of his career for the Coyotes, who racked up their fourth win in five games.
Mikkel Boedker and Keith Yandle also found the net for Phoenix.
Danny Taylor, making only his second career NHL appearance and first since 2007-08, turned aside 33-of-37 shots for the Flames.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys