The Blues won for the sixth time in eight games.
Elliott came into the game leading the NHL with a 1.52 goals-against average and tied for top spot in save percentage. He ran his record to 13-3, turning Phoenix aside after the Coyotes cut their deficit to one goal on a score by Keith Yandle with 6:21 to play.
Matt D'Agostini put St. Louis in front for good at 2-1 with 1:35 to go in the second period and Jason Arnott scored what proved to be the winning goal 1:48 into the third.
Phoenix had won its previous two games. The Blues put only 18 shots on goal for the entire game, but got three of them past Jason LaBarbera.