The Predators became the first road team to win a game in the NHL playoffs and knocked goaltender Dan Ellis out of the contest in the process.
Shea Weber scored for Nashville in the opening period and Steve Sullivan increased the lead with 4:44 to go in the second. Fisher then produced goals with 1:52 remaining in the middle period and 56 seconds into the third.
Rinne's bid for a shutout was spoiled by Teemu Selanne's power-play goal with 8:36 to go in the game.
Ellis gave up four goals on 24 shots before being replaced by Ray Emery.
The second game of the series will be played Friday in Anaheim.