Rinne, recovering from off-season hip surgery, played the opening two periods before being relieved by Marek Mazanec, who gave up a goal on three shots he faced in the third period. It was the second consecutive game that Rinne played two periods and did not allow a goal.
Moser, a Swiss native on a professional tryout agreement with the Predators, scored off a pass from Rich Clune at 4:48 of the second. and Klein made it 2-0 with 5:13 remaining in the third.
Tampa Bay avoided the shutout when Steven Stamkos scored with 46 seconds remaining. Anders Lindback, Rinne's backup in Nashville for two seasons before being traded to Tampa Bay in June 2012, had 31 saves.