The Predators have won nine of their last 11 games and they have scored at least four goals in seven of those nine victories.
Sergei Kostitsyn opened the scoring 3:15 into the game and Franson put a slap shot past Craig Anderson with 3:23 to go in the first period.
It remained a 2-0 contest until the third period, when Matt Halischuk, Legwand and Colin Wilson all scored to break things open.
The only goal off Rinne came from Matt Duchene, his 19th of the season, with 3:37 to go.
Anderson allowed five goals on 28 shots. The Avalanche had won their previous two games.