Elliott turned aside all 20 shots he saw for the Blues, who leapfrogged the New York Rangers for the most points in the league with 103 in the race for home-ice advantage throughout the playoffs.
St. Louis also established a four-point lead over idle Vancouver for the Western Conference's best record.
David Backes and Alex Pietrangelo each recorded a goal and an assist, while Kevin Shattenkirk notched three assists as the Blues won for only the second time in six games.
Jason LaBarbera made 25 saves for Phoenix, which remained tied with Dallas for the Pacific Division lead with five games remaining.
The Stars have two games in hand over the Coyotes.
2014: The Year in Music [PHOTOS]