Tarasenko's 18th goal of the season early in the third period came just 16 seconds after Roman Josi had given the Predators a 3-2 lead.
In the shootout, T.J. Oshie scored for St. Louis and Ryan Ellis responded with a score for the Predators.
Josi missed the net to begin the third round, and Tarasenko then decided it with wrist shot over the shoulder of Carter Hutton.
David Backes and Jaden Schwartz scored in regulation for St. Louis, which claimed a fourth victory in five games.
Brian Elliott ended with 28 saves.
Hutton yielded three goals on 34 shots.