The Blues extended their winning streak to a season-long five games. The fading Predators have dropped four straight and seven of their last eight.
St. Louis strengthened its chances of making the playoffs in the West, while Nashville fell deeper into last place in the Central Division.
Elliott was relatively untested en route to his second consecutive shutout. He made four saves in the first, six in the second and five in the third. Elliott has 20 career shutouts.
"I'll take that any birthday, for sure," Elliott said. "It's kind of a gritty win. The puck was bouncing, not too many pretty plays out there, but I'll take that win any day of the week."
After a scoreless first period, Steen got the game's first goal at 16:37 of the second. Carrying the puck along the boards on the left side, Steen pulled up at the top of the left circle and fired a slap shot that beat goaltender Pekka Rinne between the pads.
"In the beginning, I thought we were on the outside a little much, but we got in front and in the tough areas the longer the game went on," Steen said. "Fortunately today, one was enough."
Steen snapped a six-game goalless stretch.
"I really liked the way we played in the second and third," Blues coach Ken Hitchcock said. "I thought we were outstanding. I thought we checked hard."
The Predators are accustomed to scoreless opening periods this season, and Tuesday was the 13th time they have gone into the first intermission with the score 0-0.
"We can't sustain any offense and it is tough to get any shots," Nashville captain Shea Weber said. "Everything we've got has been from the outside and we really did not have very many good scoring chances."
Rinne made 25 saves in the loss.
"The Blues have won five in a row and only given up five goals," Nashville coach Barry Trotz said. "They're as strong defensively as anyone in the National Hockey League."
For the second time in as many games, a Predators forward made his NHL debut as Daniel Bong played after being recalled from Milwaukee of the AHL earlier in the day. Nashville has had a rash of injuries among its forward corps, causing Austin Watson to be recalled for Sunday's game in Detroit.
Nashville's lowest shot total was 13 in a Jan. 24 loss to the Blues. Rinne has started 25 consecutive games for the Predators ... St. Louis went unpenalized in the game. The Blues have not allowed a power-play goal against in the last seven games. Elliott has won four consecutive starts.
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