Sebastian Aho tied it midway through the third period for the Hurricanes. Frederik Andersen stopped 35 shots while making his first start of the postseason.
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Cal Clutterbuck scored for the Islanders and Ilya Sorokin finished with 39 saves. New York was eliminated in the first round for the first time in their last five trips to the postseason.
The Hurricanes advanced to face the winner of the series between the New York Rangers and New Jersey Devils. The Devils lead 3-2 heading into Game 6 on Saturday night.
Derek Stepan started the winning sequence in overtime when he intercepted Islanders defenseman Adam Pelech's clearing pass. Stepan skated up and passed to Stastny, who fired a sharp-angle shot from the goal line that deflected off Sorokin's left pad and between his legs.
It was Stastny's third career overtime playoff winner.
After getting outshot in each of the first two periods, the Hurricanes had a 19-5 edge in the third and tied the score midway through the period.
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Sorokin denied a backhand shot by Jesse Puljujarvi 2:13 into the third to keep the Hurricanes off the scoreboard. At the other end, Andersen saved a shot by Casey Cizikas less than 10 seconds later. Sorokin also had a nice glove save on Jordan Martinook at 8:17.
Aho from the left doorstep at 9:24 as he knocked the puck down out of the air with his glove and knocked it past Sorokin. It was his fourth goal of the series.
The Islanders outshot the Hurricanes 17-11 in a scoreless second period that saw both sides have plenty of scoring chances.
Sorokin gloved Brady Skjei's point shot 1:03 into the second period. Andersen then gloved a shot from between the circles by Brock Nelson four minutes later.
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