The Coyotes moved from a tie for seventh place with Los Angeles all the way into a tie for fourth with Chicago and Calgary in the tightly-bunched Western Conference race. Phoenix, Chicago and Calgary all have a one-point lead over Dallas and a two-point advantage over Los Angeles.
It was the 22nd career shutout for Bryzgalov, who has spent four seasons each with Anaheim and Phoenix.
The game was scoreless until Brett MacLean put a wrist shot past Henrik Karlsson 1:51 into the second period. Eric Belanger scored 4:49 into the third and Shane Doan added an empty-net goal with 1:56 remaining.
Phoenix won for just the second time in eight games. Karlsson stopped 29-of-31 shots in Calgary's first loss in four outings.