Halak started the game and made 12 saves before leaving the game early in the second period after sustaining an accidental hit from teammate Barret Jackman.
Elliot relieved and turned aside all 17 shots he saw the rest of the way for the second-seeded Blues, who won a postseason game for the first time since 2004.
David Backes, Andy McDonald Vladimir Sobotka found the net for St. Louis.
Antti Niemi made 29 saves for the Sharks.
Notable deaths of 2014 [PHOTOS]
LeBron James reveals his new Nike shoes