Elliott, making only his second start of the season, has 22 shutouts in a seven-year career spent with Ottawa, Colorado and St. Louis. This was Elliott's 99th NHL win.
St. Louis has captured three in a row and the Blues blanked Florida for the second time this year, having produced a runaway 7-0 victory over the Panthers Oct. 5.
Derek Roy scored the only goal of the first period and St. Louis produced three scores in the second. The last of those came from Steen, who has an NHL-best 12 goals for the season.
All four goals were allowed by Jacob Markstrom on 23 shots.