Rinne, in his sixth season with the Predators, agreed to a seven-year, $49 million deal earlier in the day. That makes him the highest-paid player in franchise history.
The signing came on his 29th birthday and he still managed to keep his focus by making 35 saves. He has 23 shutouts in his career and is already within four of equaling his best single-season total.
Jonathon Blum and Matt Halischuk scored in the second period and Patric Hornqvist added an empty-net goal with a minute remaining in the third.
Blum's goal was his first of the season and only the fourth he has scored in his two years with the Predators.
Jason LaBarbera stopped 29-of-31 shots for the Coyotes, who had won their previous three games.