Rinne, recovering from offseason 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.
Steven Stamkos had the lone goal for the Lightning at 19:14 of the third period.
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. Klein made it 2-0 with 5:13 remaining in the third.
Anders Lindback, Rinne's backup in Nashville for two seasons before being traded to Tampa Bay in June 2012, had 31 saves.
"It was fun to play against Lindy," Rinne said. "It was good to see him play extremely well, but we still got the win.
"Lindback saw a lot of action, but I think he played pretty under control," Lightning coach Jon Cooper said. "He made the saves and that’s what you want in your goaltender, to make all the saves you can. You can’t expect Lindback to keep making all the saves, but he was under control and played the puck well. I thought he was exceptional. Plus, he’s coming into a town that he used to play in, so there’s that added pressure and he responded."