Thornton had a goal and two assists to give him 1,410 NHL points and move past Dale Hawerchuk for 19th place in NHL history.
"Our 5-on-5 play early was good," Hurricanes coach Bill Peters said. "Our specialty teams hurt us tonight. We were minus-4 on special teams. It's tough to win when you do that."
Sebastian Aho scored two goals, and Victor Rask and Jeff Skinner each scored once for the Hurricanes (11-10-6), who have lost four of their past five games (1-2-2).
"When you go into the third with a two-goal lead, you're in great shape," Skinner said. "But that was not a good enough third period. [Ward] came up with some big saves. He's probably the difference in going overtime."
Couture cut Carolina's lead to 4-3 at 8:31 in the third, and Goodrow tied it 4-4 on a breakaway at 11:56.
"Pretty good game for the fans probably, huh?" Burns said. "It's hockey. Just the last swings. That game had everything in it. Tough start, great finish. You got to try to learn what we were struggling with and it worked at the end. It felt good."
Aho put the Hurricanes ahead 1-0 at 9:01 of the first period when he took a pass from Noah Hanifin and scored on a wrist shot from close range.