Aleksander Barkov scored, and Alex Lyon made 30 saves for the Panthers (42-32-8), who had their seven-game point streak end (6-0-1) and finished as the second wild card in the East, one point behind the Islanders.
Florida will face the Boston Bruins in the first round.
"We had a lot of fight but made mistakes, which will not be allowed or acceptable in the playoffs," Panthers forward Matthew Tkachuk said. "We had to be patient in a game like that and it got away from us in a few shifts. They capitalized. [They're] a good team and the top team in their division. But the fight in the third shows how we fought all year, especially in the second half."
Burns and Jesper Fast scored 21 seconds apart in the third period after the Panthers erased a 2-0 deficit with two goals in the first 3:33 of the period.
Burns gave Carolina a 3-2 lead at 6:07 with a shot from the point through traffic. Fast made it 4-2 at 6:28, and Gostisbehere scored an empty-net goal at 17:20 to give the Hurricanes a 5-2 lead.
Burns, a 38-year-old defenseman, finished the regular season with 61 points (18 goals, 43 assists) in 82 games after being acquired in a trade with the San Jose Sharks last summer.
"I don't know where we'd be without him, to be honest," Brind'Amour said of Burns. "Our [defensemen] are special back there. And the amount of pressure we put on him to perform in every aspect of the game, and this guy is better than advertised or as good or whatever. And a real pleasure to be around and have him with our group."