Kakko gave New York a 2-1 lead 31 seconds later at 10:20, scoring from between the circles off a rebound of Ryan Lindgren's shot that was blocked by Burns.
But Noesen scored just 18 seconds later, tying the game 2-2 at 10:38 with a one-timer from the slot off a pass from Jesper Fast out of the right corner.
"We're down, tie it up and they get one back right away, I just loved the way we were just like, 'Well, next shift,'" Brind'Amour said. "Obviously, it was great to tie it up and get that late one. We got what we deserved tonight for sure."
Motte gave the Rangers a 1-0 lead at 17:00 of the first period on a one-timer from the slot off a pass from Barclay Goodrow, who was behind the left circle.
But New York could not convert on two power plays in the second period and one at 7:22 of the third before Chatfield scored to tie the game.
"We understand this time of year and moving forward it gets tougher and tougher, and you've got to have the ability to play that way," Brind'Amour said. "Obviously, you want to be able to score nine goals too. I mean, that'd be nice. But you have to do what you've got to do to your identity of your team, and tonight I think we played to that."
NOTES: Carolina has 10th third-period comeback wins this season, tied with the Vancouver Canucks and Vegas Golden Knights for most in the NHL … It was Teravainen's second game-winning goal of the season (Jan. 19, 5-2 victory against the Minnesota Wild). … Lindgren had an assist and played 16:13 in his return after missing 11 straight games with an upper-body injury he sustained Feb. 25.