The Blues inched closer to a playoff spot, building a three-point lead over Minnesota while holding down the No. 6 seed in the Western Conference.
Dallas missed out on a chance to move into a playoff position, remaining tied with Detroit two points behind Columbus for eighth place.
Chris Stewart gave the Blues the lead midway through the first period and Ryan Reaves produced what turned out to be the game-winner 7:52 into the second.
Elliott's bid for a shutout was spoiled by Antoine Rossel early in the third. The Stars lost for just the second time in eight games.
Kari Lehtonen allowed the two St. Louis goals on 20 shots.