The Blues won their third in a row in the 10th shutout of Elliott's four-year career.
Elliott was obtained from Colorado this season as a backup to Jaroslav Halak, but has been given more playing time than expected because of Halak's early-season struggles.
Halak has a 1-4 record and Elliott is unbeaten in his four starts. Elliott made 30 saves against the defending Western Conference champions, who dropped their second straight.
Steen opened the scoring 8:29 into the second period and then added a goal 6:17 into the third. The goals were his fourth and fifth of the season.
Patric Berglund ended the scoring for the Blues, who have played six of their first nine games on the road.
Cory Schneider, who had a 1.67 goals-against average in his previous four appearances, stopped 28-of-31 shots to take the loss.