Phoenix won its eighth straight home game, jumping to a 2-0 lead after one period on goals by Scottie Upshall and Martin Hanzal.
Anaheim's Jonas Hiller allowed three goals on 28 shots before being replaced by Jean-Sebastien Giguere midway through the third period. The Ducks had won their previous two games.
The shutout was the 13th in Bryzgalov's seven-year career. Prior to this season, his single-year shutout high had been three.
Hanzal's goal in the first period brought him the 200th point of his career.
2014: The Year in Fashion [PHOTOS]
LeBron James reveals his new Nike shoes