Elias Lindholm scored the other goal while Cam Ward made 24 saves for the Hurricanes, who had dropped six of seven coming into the contest.
"The big word on sports teams is chemistry. For us to move forward here from now to the end, your best players have to come up big. He's one of the leaders of this team,," said Carolina head coach Kirk Muller about Ward. "I think he wanted it. He's looked more comfortable in the last three or four days."
"We gave away two points because we didn't work," said New York head coach Alain Vigneault. "Right from the start our work ethic wasn't what it should be in any game in this league if you want to win. We'll deal with it tomorrow and go from there."
With the score tied in the second, Carolina went on top as Staal fought for the puck at the right boards and fired it over to the left circle for a wide open Tlusty, who sent it back to the right where Staal shot it home for a 2-1 lead with 1:42 to play in the second.
The Hurricanes put the game away with 5:29 to play in the third after Jeff Skinner's shot from the low right circle was stopped and Lindholm was in front to backhand home the rebound.
Carolina drew first blood just 33 seconds into the second period when Staal sent a short pass from the low left circle to the front and Tlusty one-timed it home.
New York, though, answered four minutes later after Mats Zuccarello sent a centering pass from behind the net and Klein wristed it in for his second of the season.
Lundqvist turned away 13 shots in the first frame while Ward needed to make just five stops to keep the game scoreless.