Koskinen, who lost his first career start Tuesday in Toronto, turned aside Montreal's first four shots in the tiebreaker after stopping 37 shots during the game.
The Islanders failed to convert their first three attempts before Okposo shot past Alex Auld to hand the Canadiens' their third straight loss.
Michael Grabner drew the Islanders even at 3 with his second goal of the game 4:08 into the third. Grabner's first goal of the game 17:10 into the second tied it at 2.
John Tavares scored his 19th goal in the first as New York erased three one-goal deficits. Frans Nielsen and Jack Hillen each assisted on both of Grabner's goals.
Max Pacioretty, Benoit Pouliot and Tomas Plekanec scored for Montreal.
Auld made 30 saves in his eighth start.
Matt Moulson hit the left post with 41.2 seconds left in the second after he jumped on a Montreal turnover in the slot. He missed another chance to give the Islanders their first lead at 12:45 when he put a backhand wide of the right post after receiving P.A. Parenteau's return pass on a 2-on-1 break.
Pacioretty scored his third goal in two games 14:10 in to put the Canadiens up 1-0.
Tavares drew the Islanders even at 16:35. The center snapped a shot past Auld for his 19th goal.
Montreal went up 2-1 when Pouliot got his first point in eight games with his 11th goal 5:49 into the second.
The Islanders drew even again at 2 when Grabner drove the slot between Canadiens defensemen P.K. Subban and Jaroslav Spacek and beat Auld between the pads with a shot along the ice for his 18th goal.
Plekanec scored his 19th goal at 18:08, shooting past Koskinen on a 2-on-1 break with Andrei Kostitsyn to put Montreal up 3-2.
Grabner tied it at 3 with his second of the game early in the third, tying Tavares and Moulson with his 19th goal.
Notes: G Joel Martin, who has yet to play an NHL game, backed up Koskinen for a second straight game. He was recalled on an emergency basis from Bridgeport of the AHL on Tuesday. ... Canadiens D Hal Gill did not play because of an upper-body injury. ... Forward Mathieu Darche did not dress for Montreal.