Roope Hintz had a goal and an assist, and Anton Khudobin made 18 saves for the Stars (15-14-11), who had won four of six.
"We played great," Stars coach Rick Bowness said. "We played our butts off. Everyone did, for the fourth game in six nights. And to dominate the third period like we did, it's disappointing just to get into overtime. But everyone on that team tonight, they worked their butts off. We played a really, really solid game."
Hintz gave the Stars a 1-0 lead at 4:56 of the first period on a shot from the slot. John Klingberg knocked a puck down at the blue line to keep it in and found Hintz in the slot on a backhand pass.
"The more emotion there is out there, the more fun it is to play," Klingberg said. "It's obviously, I wouldn't say easier to motivate, but it's a lot more fun when it's a grinding game like that. It's huge points and these games are the ones that matter. That's the ones you want to play."
Ellis tied it 1-1 at 10:57 on a wrist shot from the point. It was his second game after missing 20 games with an upper-body injury.
"We really focused on the competitiveness of the game," Predators coach John Hynes said. "We're all aware of the standings. It's the best time of year, and this is when the most competitive games have got to bring out the most competitive men.
"We talked about really enjoying the process. We have a lot of respect for Dallas. They're a good team. They're a hard team to play against. They're really well coached, so we knew coming in, we talked about it this morning, it's going to be Masters Sunday and it was going to be a playoff-type game for us. Just really proud of the way the guys have readied themselves for the games."