Seth Jarvis, who had two assists, scored an empty-net goal with 1:39 remaining for the 6-3 final.
"I mean, that's just a tough way to lose a game," Islanders captain Anders Lee said. "We were grinding back, stayed in the fight all night. We believed what we were doing, believed we were going to win this hockey game, and two bounces like that that. ... It's tough to swallow, but that's how it goes sometimes."
Evgeny Kuznetsov had a goal and an assist, and Frederik Andersen made 22 saves for the Hurricanes, who are the No. 2 seed from the Metropolitan Division.
Carolina will play the New York Rangers, who swept the Washington Capitals, in the second round.
“The start of the game was great. That at least got us a nice lead,” Hurricanes coach Rod Brind’Amour said. “They had their push, and you’ve got to give [the Islanders] a lot of credit. They just kept coming. We had to play really well to win this series.
"It’s hard to win a playoff round. Sometimes you forget that. I’m proud of the group because that’s a good accomplishment, but that’s not what we’re shooting for.”
Mike Reilly, Brock Nelson and Casey Cizikas scored for the Islanders, who were the No. 3 seed from the Metropolitan. Varlamov made 32 saves.
"I am very proud of this group because they were resilient,” New York coach Patrick Roy said. “I mean, even tonight they show that. That's how they were all year. I mean, it would have been easy to pack it in, and it's not what we did.
"I feel like we deserve better. I thought we played some good hockey in that series. Losing in five, I feel like we deserve a little bit better, and I'm not saying we should have won that series, (but) I'm saying that we could go home right now and play game No. 6 easily, and instead it's over."
Carolina took a 1-0 lead on its first shot of the game at 1:23 of the first period. Teuvo Teravainen took a pass from Jarvis and scored under the left arm of Varlamov from a sharp angle.