With the shootout tied after five rounds, Tim Thomas turned aside Gabriel Bourque and Kopecky followed by firing a shot off the far post that went past Mazanec for the game-winner.
Kopecky also scored in regulation and Thomas stopped 38-of-42 shots for Florida, which completed a 2-2-1 homestand.
Mazanec logged 25 saves.
Ray Liotta sues skin care company over use of likeness
NBC reportedly holds celebs hostage to Jimmy Fallon's show