Pekka Rinne made 27 saves for the Predators (14-17-1), who have won three of their past four games and went 4-3-1 on an eight-game road trip, the longest in their history.
"I'm ready to get in bed," Nashville forward Ryan Johansen said. "I'm exhausted. But you know, we found a way to win some games and get some points on this trip against some great hockey clubs. Overall, we're happy with this trip, especially with all of our guys out of the lineup, as everyone knows. Looking forward to getting back home and getting in bed."
The Stars (10-10-8) have lost three of four (1-1-2).
Dallas forward Radek Faksa left the game after blocking a shot in the second period, and forward Joel Kiviranta left after one shift in the third period because of a lower-body injury.
"I think we battled hard, but obviously that hurts," Stars forward Ty Dellandrea said. "Going pretty well three lines wears you down, but we had the same schedule as they did and both are tough. It's not an excuse at all, but it definitely hurt."
Each team was playing the second of back-to-back games. The Predators lost 2-0 at the Florida Panthers on Saturday, when the Stars won 3-0 at the Detroit Red Wings.
Calle Jarnkrok scored the deciding goal in the fifth round of the shootout. Johansen scored in the bottom of the second round, and Dallas' Joe Pavelski scored in the top of the third.
Tolvanen tied the game 3-3 at 14:15 of the third period on a shot through traffic from above the right face-off circle.