Steen's power-play tally 5:27 into the game put the Blues ahead to stay and his second goal of the game in third gave St. Louis a 4-1 edge.
His second goal tied him with Washington's Alex Ovechkin for the NHL lead.
T.J. Oshie, Alex Pietrangelo and Jaden Schwartz each provided a goal and an assist and Derek Roy also scored for the Blues, who improved to 6-1-2 on the season.
Jaroslav Halak stopped 22-of-23 shots.
Carter Hutton started in goal for Nashville but was pulled in the third period after yielding five goals on 26 shots. Magnus Hellberg saved 3-of-4 St. Louis tries the rest of the way.
Nick Spaling scored for the Predators, who were coming off a Friday overtime win against Winnipeg.