"Tarasenko is dangerous off the rush," Trotz said.
Derek Roy, who played on a line with Tarasenko and Jaden Schwartz, said the sky is the limit for Tarasenko.
"I feel like he gets better every game," Roy said. "He's eager to learn. He wants to play well and he wants to win. Those are good qualities in a player."
Schwartz opened the scoring with a power-play goal at 9:11 of the first period when he converted a centering pass from Chris Stewart. Backes gave St. Louis a 2-0 lead when he redirected Jay Bouwmeester's shot from the right point at 3:31 of the second.
But Gaustad got the Predators back in it with 3:12 left in the second when the puck came to him in the slot off a scrum in the corner, and he buried a shot past Elliott.
NOTES: Nashville D Shea Weber, who scored twice in Friday night's 3-2 win against New Jersey, was listed as questionable because of an upper-body injury but he played his usual shifts. ... Backes played in his 543rd game with the Blues, tying him with Keith Tkachuk for 12th place on the team's career list. ... Josi has 11 points in his last 10 games.