Evgeni Nabokov made 20 saves in the win. Eddie Lack stopped 21 shots for the Canucks.
The Islanders let two-goal leads get away in the third periods of losses to Edmonton and Calgary on consecutive nights last week.
New York went 2-1-1 on its four-game trip and improved to 8-22-2 when trailing after two periods this season. The Islanders are only 9-6-5 when leading after two.
Vancouver was ahead 1-0 after the first period and 3-0 following the second before allowing the Islanders back in it.
New York's first three goals came on power plays, and Nielsen's goal came right after another Vancouver penalty had expired.
Higgins opened the scoring on a power play at 15:16 when he put in a backhand after Yannick Weber's slap shot hit the post.
Henrik Sedin put the Canucks ahead 2-0 at 4:52 of the second as he put a shot in off Clutterbuck from the behind the net for his first goal in 24 games.
Canucks rookie Nicklas Jensen, recently called up from the AHL, earned his first NHL point with an assist.
Kesler made it 3-0 about five minutes later when he put in Kevin Bieksa's rebound off the goal post. Alex Burrows earned an assist for his 300th NHL point.
Bailey started the comeback at 1:13 of the third while Jensen was serving a slashing penalty that carried over from the second period.
Strome and de Haan scored power-play goals 22 seconds apart to tie it.
Nielsen put the Islanders ahead 4-3 just under three minutes later with a shot from the wing.
The crowd let out a huge roar of relief when Tanev tied it 4-4 at 9:54.
Martin, however, gave the Isles the lead only 10 seconds later on a backhand from the slot, and Lee padded New York's lead 1:44 after that.
Lee put in Okposo's rebound off defenseman Dan Hamhuis, who had tended the goal while Lack was caught out of position following the initial save on the first of three New York shots.
Clutterbuck added an empty-net goal in the final minute.
NOTES: The Islanders had seven rookies in their lineup. . . . New York lost forward Michael Grabner late in the second period after he collided with Tanev. . . . Canucks forward Zack Kassian served the second of a three-game suspension. . . . Islanders D Thomas Hickey is the only player to dress for all of New York's 67 games this season.