Elliott came into the game with the league's best goals-against average of 1.63 and helped lift the Blues within three points of Detroit for the best record in the Western Conference.
Jason Arnott tied it with 5:20 to go in the first period and Kevin Shattenkirk put the Blues in front by scoring a power-play goal with 2:48 remaining.
St. Louis' other goals came in the second period from Andy McDonald, David Perron and T.J. Oshie. McDonald, who has missed almost all of the season due to a concussion, scored his first goal of the year.
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