Vrbata delivered the game's final three goals, starting his big night by breaking a 1-1 tie 7:05 into the second period.
He added two goals in the first 8 minutes of the third, the first of those coming on the power play.
It was a solid season opener for the Coyotes, who finished last in the Pacific Division during the lockout-shortened 2012-13 season.
Smith made 23 saves in picking up the victory. The loss went to Henrik Lundqvist, who had 28 saves.
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