It was the 25th career shutout for the veteran Smith, who played in his 303rd NHL game.
The game's only goal came from Antoine Vermette with 7:09 to go in the first period. Vermette tallied while Vancouver's Jason Garrison was in the penalty box for hooking.
Phoenix snapped a four-game losing streak and allowed the Coyotes to move within four points of Vancouver for fourth place in the Pacific Division.
Eddie Lack suffered the hard-luck loss while making 22 saves for the Canucks.