Miikka Salomaki, James Neal and Shea Weber also scored for Nashville.
Playing in just his ninth game in place of starter Pekka Rinne, Hutton ran his record to 6-2-1 and picked up his second shutout.
Florida dropped the first two games of its six-game homestand, which it continues Monday against Pittsburgh.
Panthers goaltender Roberto Luongo made 23 saves in his second straight loss.
Both teams were coming off losses Friday night.
Mike Ribeiro and Roman Josi had two assists each, and Neal, Austin Watson, Anthony Bitetto, Ryan Johansen and Roman Josi had one each.
