The last place team in the Metropolitan Division defeated the defending Stanley Cup champions after letting an early 2-0 lead slip away.
New York managed only nine victories in its first 35 games this season, but has now run off five victories in seven contests.
Casey Cizikas and Thomas Vanek gave the Islanders their two-goal lead midway through the game, but Brent Seabrook scored with 1:24 to go in the middle period and Ben Smith tied it 1:50 into the third.
Okposo brought a quick end to the overtime by putting a wrist shot past Corey Crawford for his 16th goal of the season.
Crawford allowed three goals on 34 shots while Evgeni Nabokov made 35 saves to collect the victory.