Curtis McElhinney made 23 saves in his season debut for the Lightning (8-1-1), who will the play the Predators here again Tuesday.
"I think it was an ideal start for our team," Lightning coach Jon Cooper said. "To have as little work as [McElhinney] had in the first period (six shots) was probably good for him. And then we settled in as a team, and the only thing that really hurt us in this game was taking the penalties (3-for-4 on penalty kill)."
Dante Fabbro scored, and Pekka Rinne made 31 saves for Nashville (5-7-0), which has lost four of its past five games and seven of its past 10.
"It's frustrating for sure," Predators captain Roman Josi said. "It's such a short season and it's important to get off to a good start. Playing in your division the whole year, consistency is a huge thing. Like I said, that's something we need to figure out and we need to figure out quickly. It's got to come from us. We need to figure it out. We don't have any time to play inconsistently. We've got to find a solution quick."
Cirelli gave Tampa Bay a 1-0 lead at 9:07 of the first period, scoring on a sharp-angled slap shot from along the left boards.
Cirelli made it 2-0 at 13:02 of the second period when he scored on a rebound after Mikhail Sergachev's initial attempt hit the post on the power play.
"He also has to play against pretty formidable lines every single night," Cooper said. "He's spent a little bit of time in the League and now the offensive side of the game is coming. I think he's got a little more confidence on the power play and, I don't know, getting rewarded for playing a great 200-foot game. It's just a matter of time."