Stamkos scored on a one-timer from the high slot following a high-sticking penalty on Mattias Ekholm.
"It was one of those games where both goaltenders were playing really well," said Stamkos, who has scored in three straight games. "We had some really good chances early in the game, their guy made some good saves, [Andrei Vasilevskiy] made some big saves. … Just create an opportunity in overtime. If you get a power play, you want to bury it. We did that. It was a just a good character win for our group."
Stamkos' goal came after Alex Killorn tied it 2-2 at 16:39 of the third period, redirecting Ian Cole's shot from the point.
"You just can't force it," Killorn said. "[Juuse Saros] made a ton of great saves. He was really good for them tonight. Once we got one … it was just continue, continue, continue. If he's going to make that many saves the entire game, good on him, but we were able to get a couple in at the end."
Nikita Kucherov had a goal and two assists, and Vasilevskiy made 30 saves for the Lightning, (11-6-1), who extended their winning streak to a season-high four games.
Saros made 36 saves, and Filip Forsberg and Nino Niederreiter scored for the Predators (8-8-2), who had won three straight.
"At the end of the day, you're that close to getting two points against a really good team," Forsberg said. "I thought we battled hard. It feels like we lost a point, but at the same time, I feel like our effort was solid. At this point of the season, we've got to take those things, too."