Hutton, who has seen plenty of action while regular Nashville goaltender Pekka Rinne has been recovering from a severe hip infection, gave up four goals on 14 shots before being replaced by Marek Mazanec.
Alex Killorn and Martin St. Louis delivered first-period goals to put Tampa Bay in charge and the Lightning went on to win their third in a row.
Nashville twice moved within a goal in the second period, but each time the Lightning responded on scores from Teddy Purcell and Richard Panik.
Anders Lindback made 29 saves for Tampa Bay. Nashville lost its second straight.