The loss hurt Phoenix's quest for the Western Conference's eighth and final playoff spot. The Coyotes entered the day two points behind the Dallas Stars in the wild card race. Both teams have two games remaining, including Sunday's season finale when Dallas visits Phoenix.
Phoenix will need to finish the season at least one point ahead of Dallas in the standings since the Stars will hold the first tiebreaker of more regulation and overtime wins no matter what happens over the course of the final two games.
Smith scored the game's first goal at 10:26 of the second period. After taking a pass from defenseman Ryan Ellis above the left circle, Smith skated down to the left faceoff dot, where he used two Coyotes standing in front of goaltender Thomas Greiss as screene and scored his 21st goal of the season with a wrist shot high to the far side.
Smith's power-play score snapped a 13-game goal-scoring drought.
Hornqvist doubled the Nashville lead at 9:21 of the third. From below the goal line, Mike Fisher found Hornqvist in the slot. Greiss denied Hornqvist's first attempt, but he poked in his own rebound for his 21st of the season - and seventh goal in his last six games.
The shutout was Rinne's second of the season and 32nd of his career. He now has 162 wins, moving him past Tomas Vokoun into first place on Nashville's career wins list.
NOTES: Prior to the game, the Predators announced that they are partnering with former Olympic figure skating gold medalist Scott Hamilton, who will serve as an active principal for the skating center at their new suburban rink set to open this summer. Hamilton is a Predators season ticket holder. ... Phoenix RW Martin Erat played in his 800th career NHL game. His first 723 were played with Nashville prior to his trade to Washington at last season's trade deadline. ... Six of Smith's goals this season have come on the power play.