Smith turned aside all 10 shots he faced in the second period as the Coyotes were scoring three times to break the contest open en route to their third consecutive win.
Smith's shutout was the 24th of his career and tied him with Nashville's Pekka Rinne for the most this season.
Oliver Ekman-Larsson helped out on goals by Shane Doan and Mikkel Boedker. Martin Hanzal and Boyd Gordon also scored for Phoenix, which pulled within two points of the Western Conference's eighth and final post-season slot.
Semyon Varlamov made 24 saves for the Avalanche, losers of four straight.