The win moved the Predators past the Blues into first place in the Central Division. The Predators lead the Blues by one point but also have played three fewer games.
The Blues fell behind 2-0 after the first two periods before ruining Rinne's bid for his fourth shutout of the season on a goal by Vladimir Sobotka, who deflected a shot by Kyle Brodziak with 6:23 to go in the third period.
Rinne shut out the Blues 2-0 on Nov. 24 in St. Louis in the only previous meeting this season.
The win snapped an 0-2-1 streak by the Predators in their last three games before the Christmas break, their longest slide of the year.
Smith's 13th goal of the season came at 10:37 of the first period when he deflected a shot by Roman Josi past goalie Jake Allen.
Rinne protected the 1-0 lead until Jarnkrok scored at 16:49 of the second period, capitalizing on a Blues' turnover.
The Blues fell to 2-6 in their last eight games as they continued to have problems scoring at home. They have scored a combined four goals in their last four home games.
The Blues just missed scoring right at the start of the game, when a shot by defenseman Vince Dunn deflected off the post one minute into the game. Rinne also robbed Brayden Schenn with a big save early in the third period.
NOTES: The Blues on Wednesday announced a four-year contract extension for Doug Armstrong, the team's president of hockey operations and general manager. The new deal also includes an option for a fifth year for Armstrong, who has been the GM since 2010. ... LW Filip Forsberg played in his 300th career game for the Predators. ... The Blues will play at Dallas on Friday night while the Predators begin a home-and-home series with the Minnesota Wild the same night.