Cal Clutterbuck, Matt Martin, Ryan Strome and Calvin de Haan each scored in the final frame for the Islanders, who had dropped four of five coming in. Evgeni Nabokov gave up four goals on 24 shots in the win.
"We should have had this game," said Sedin. "We played great hockey for 37 minutes but then we started making poor decisions. We should have been back in the race but now we're down and looking up."
The Islanders trailed, 3-0, entering the third period, but were skating on a power play and got one in as Bailey wristed the puck in from the right circle at 1:13.
A pair of penalties to Vancouver saw New York skating 5-on-3 and the team quickly made it a 3-2 game as passing around the horn was finished off on a one-timer by Strome at the low left side at 3:14.
Still on the advantage, the Isles tied the game 22 seconds later when de Haan's shot from the left point was knicked by a defender and sailed past Lack.
New York then took its first lead of the game three minutes later, when Kyle Okposo's shot from the slot hit off Lee and went to Nielsen, who put it in for his 20th of the season.
Vancouver tied the game at 9:54 as a long wrist shot from the right point by Tanev went in, but New York took the lead back just 10 seconds later as Lack stopped a dump-in behind his net where Colin McDonald picked up the puck and centered it for a one-timer by Martin.
The Islanders got an insurance goal less than two minutes later as a 2-on-1 break saw Lack make two nice saves, but the third chance went in as Lee wristed it home from the left circle.
Clutterbuck scored with the net empty inside the final minute to seal the victory.
"It does feel good to put seven on the board and come back in the fashion that we did," said Islanders defenseman Travis Hamonic.
Higgins scored on the power play at 15:46 of the first period to get Vancouver on the board, while Sedin and Kesler added markers four minutes apart in the second, with Kesler's coming on the power play, for a 3-0 lead.