Jamie Benn had a goal and an assist, and Anton Khudobin made 37 saves for the Stars (4-1-1), who lost 4-1 to the Hurricanes on Saturday. Alexander Radulov scored in the shootout.
"The result is obviously not what we wanted," Benn said. "They're a good team. They came out flying. I thought we got better as the game went on. We're right on the brink of being really dangerous."
Benn scored on a wraparound at 12:00 of the third period to give Dallas a 3-2 lead. He missed the previous four games with a lower-body injury.
Staal scored on the power play to give the Hurricanes a 1-0 lead at 1:58 of the first period when he screened Khudobin in the low slot and tipped Hamilton's one-timer for his first goal of the season.
Jamie Oleksiak tied it 1-1 when he put in the rebound of Benn's shot at 7:47 of the second period.
"In the second period, 5-on-5, we started carrying the play," Stars coach Rick Bowness said. "So that looked a lot more like the Dallas Stars. We had some great opportunities, a couple of posts, and that carried over into the third."
Brock McGinn put Carolina ahead 2-1 with a shorthanded goal at 16:32. Staal forced a turnover in the neutral zone and fed a backhand pass to McGinn for a shot under Khudobin's right pad.