The stunning turn of events gave the Hurricanes a 4-1 series win.
It came after the Islanders, playing from behind all night, tied it at 3 late in the second period.
But the goals from Jack Drury and Stefan Noesen in quick succession sent PNC Arena into a frenzy and the Islanders home for vacation.
The Hurricanes started quickly in Game 5, getting a goal just 1:23 into the game from Teuvo Teravainen. They padded the lead to 2-0 less than two minutes later when Andrei Svechnikov scored on a power play. Evgeny Kuznetsov added a slow-play score on a penalty shot before the end of the period for a 3-1 lead.
But Carolina twice blew two-goal leads and entered the final 20 minutes tied before Drury struck for his first career postseason goal, controlling a dribbling puck to zip it past Semyon Varlamov to his blocker side at 4:36 of the third.
Then, after a faceoff win, the Hurricanes dumped the puck into the Islanders' end toward the corner. As Varlamov went behind the net to play the puck, it took an unexpected bounce near the door and caromed right into the crease.
Noesen charged in to bury it as Varlamov tried desperately to get back to the netfront, pushing Carolina to a 5-3 lead at 4:44.
Ultimately, that was enough to help the Hurricanes finally push past the determined Islanders and put the series on ice. They also earned at least one playoff series win for the sixth time in as many seasons. And that earned them a date with the Presidents' Trophy-winning New York Rangers in the second round.
Carolina jumped to a 3-0 lead in this best-of-7 series before missing out on a chance to clinch in Saturday's double-overtime road loss. That set up a familiar scenario from last year when the Islanders won Game 5 here to extend that first-round series before falling in six games.
This time, Carolina closed it out even after a tense vibe in its typically rowdy home arena going into those final 20 minutes. By the end, though, Seth Jarvis had added an empty-net clincher at the 18:21 mark to let Hurricanes fans stay in a celebratory roar to close this one out.
Noesen's bizarre goal captured some of the wild action, which included New York's Casey Cizikas scoring in the final seconds of the second on an unguarded net - which emerged when Carolina goaltender Frederik Andersen simply stumbled as he scrambled to his right after a stop and fell untouched out of the crease.
Mike Reilly and Brock Nelson also scored for the Islanders, who won eight of their last nine games to clinch a playoff bid in the waning days of the regular season. That came after a January coaching change with the firing of Lane Lambert to hire Patrick Roy.
Carolina entered the playoffs as the favorite to win the Stanley Cup according to Bet MGM Sportsbook, but the Islanders gave the Hurricanes fits the entire way. That included outplaying Carolina for much of the Game 1 loss, then blowing a 3-0 lead by giving up the tying and go-ahead goals 9 seconds apart in the final 3 minutes of Game 2.
Ultimately, another improbably quick burst helped finish off the Islanders.
Varlamov had 32 saves, while Andersen finished with 22.
The Associated Press contributed.