Tolvanen got past forward Denis Gurianov, received a pass from Viktor Arvidsson, and beat goalie Anton Khudobin with a backhand on a breakaway.
"I don't know. I don't know. That was a gift," Dallas coach Rick Bowness said of Gurianov's backcheck on the play. "That's all I'm going to say on that. That was a complete gift."
Juuse Saros made 33 saves for the Predators (19-17-1), who have won eight of their past nine games.
"Today was probably one of the most physical games all year," Predators captain Roman Josi said. "I think every time you play Dallas, they're a hard team to play against. I think it shows what our team can do in games like that, and it's definitely a huge win for us."
The Stars (11-12-10) have lost three games in a row and are 1-2-3 in their past six.
"It's hard," Dallas defenseman Mark Pysyk said. "We've been in a lot of overtime games that haven't gone our way. We're happy with the point, but we definitely wanted two."
The teams will play again here Thursday.
Calle Jarnkrok tied the game 2-2 for the Predators at 4:05 of the third period when he scored from the left of the net after Luke Kunin's shot went wide and caromed off the end boards.
"I thought it was two teams who played very well today," Jarnkrok said. "I think they might've been a little better than us in the first and the second periods. We just kept going and had our best period last, and it worked today, and we stayed with it the whole game. It was a big win for us."
Jason Robertson gave the Stars a 1-0 lead at 6:12 of the second period on a wrist shot from the left face-off circle off of a backhand pass on the rush from Gurianov.
Josi tied it 1-1 at 9:04 with a one-timer on the power play. It was his 434th NHL point, tying defenseman Mark Streit for the most by a Switzerland-born player.
"It's a huge honor," Josi said. "Me and [Streit] are really good friends. He did so much for Swiss hockey because we never had a player that really made it in the NHL. We had a couple of goalies, but he was the first guy really who made it and had a big role on the team. He was a star in the League. For us as young kids, we all looked up to him, and he kind of gave us that belief that it's possible for Swiss players to make it."