Anton Khudobin made 33 saves for Dallas (21-17-13), which has lost three in a row.
"We played our hearts out," Stars coach Rick Bowness said. "We did. We did. We didn't give up much, and obviously we didn't score, and we need to score. But in terms of the effort and the commitment to the team, 100 percent satisfied."
Haula put in a loose puck in front after a wraparound by Mattias Ekholm.
"I feel like our group felt confident," Haula said. "I think it showed with the way that we played in the third period. I thought we were dictating. We were getting the puck in their zone. We had them hemmed a couple of times for some long shifts. I think you've got to be pretty proud of the way we showed up in the third and played. Obviously it's a big game and we knew what was at stake. Being able to play your best in the third was great to see, I thought."
The Stars played their second of seven straight road games to end the regular season. They lost 3-0 at the Tampa Bay Lightning on Thursday.
"It's a tough one," Dallas forward Radek Faksa said. "I think we battled hard the whole 60 minutes. Even in the overtime we did a pretty good job on the [penalty kill]. It's a tough one, a tough bounce. We were hoping we'd win this one because it was really important. I think we worked really hard. Tough bounce, but we've still got five games to go and we've still got a good chance to make the playoffs. We need to keep the head up and keep working hard."
Nashville finished the season series against Dallas 5-3-0, with each of the wins after regulation: three in the shootout and two in overtime.
"It's big points and a big win," Saros said. "It was a really tight game. We just stayed with it and eventually got it done. … We've still got some business to do here. It's a fun, tight race."
NOTES: Predators forward Filip Forsberg had three shots and played 18:20 in his return after missing 17 games because of an upper-body injury. … Nashville defenseman Alexandre Carrier played 21:14 in his first game after missing 13 because of a lower-body injury. … Predators forward Brad Richardson had two shots in 10:59 returning to the lineup after missing 25 games because of a lower-body injury. … Nashville defenseman Roman Josi left the game in the first period after his head hit the ice after a hit into boards by Dallas forward Jason Robertson but returned in the second period. He missed 17:32. … Stars forward Jason Dickinson did not play in the third period because of a lower-body injury, and forward Denis Gurianov was a late scratch with an upper-body injury. There was no update. ... The Carolina Hurricanes, Florida Panthers and Lightning have qualified for the playoffs in the Central.
