WATCH: [All Predators vs. Stars highlights]
Alexander Radulov and Tyler Seguin scored in the shootout, and Ben Bishop made 29 saves to help Dallas (20-14-3) win back-to-back games and extend a four-game point streak.
"That's great whenever you get a division game and win, that's pretty much four points, and that's [going to] build great confidence for us right before Christmas," Stars center Martin Hanzal said. "… So it's great for our group and we've got to build on that."
Bishop stopped Filip Forsberg and Kevin Fiala in the shootout before Seguin scored in the third round.
Juuse Saros made 25 saves for Nashville (21-9-5), which led 2-0 but lost its third straight game.
Ryan Johansen scored for the Predators to tie the game 3-3 with a power-play goal at 4:45 of the third period.
Stars forward Jamie Benn was called for tripping at 1:05 of overtime and defenseman Dan Hamhuis cross-checked Filip Forsberg at 2:34.
"I take a lot of responsibility myself. We have that long a time to play 4-on-3 and even 5-on-3, we've just got to find a way to score," Predators defenseman Roman Josi said. "Us guys that are out there, we've just got to score. That much time, you [have to] put the game away."
The Stars won four of the five face-offs while shorthanded, and Bishop made five saves in overtime.
"It's huge, our goalie made some good saves, our penalty killers stepped up and succeeded with the pressure against from a great power play," Benn said. "We did a hell of a job tonight in that area."
Nashville took a 1-0 lead on its first shot 34 seconds into the first period. Alexei Emelin scored his first goal of the season on wrist shot from the slot that beat Bishop through a screen.
Calle Jarnkrok gave the Predators a 2-0 lead at 7:03 on a shorthanded wrist shot. Dallas made it 2-1 at 12:13 when Seguin scored on a power play after a pass from Benn.