Elliott tied Jonathan Quick of Los Angeles for the league lead in shutouts and registered the 13th whitewash of his career as the Blues won their third straight and 11th in 16 games.
Kevin Shattenkirk got the game's only goal on power play with 26 seconds left in the first period.
Antti Niemi turned aside 18-of-19 shots in the tough-luck loss for the Sharks, who fell for just the third time in 14 visits to St. Louis.
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