Nielsen bested the Red Wings' Jonas Gustavsson on the first shot of the tiebreaker, and New York's Kevin Poulin -- who replaced the injured Evgeni Nabokov in the first period -- stopped Henrik Zetterberg and Pavel Datsyuk.
Todd Bertuzzi was wide with a shot on Detroit's final try, sending the Islanders to their fifth straight win over the Red Wings.
John Tavares had a goal and an assist.
Nabokov left with a groin injury when Darren Helm scored 4:12 into the game on Detroit's first shot of the game.
Poulin finished with 19 saves.
Datsyuk scored twice and Johan Franzen also had a goal as the Wings dropped a sixth straight overall.
Gustavsson stopped 37 shots.
Notable deaths of 2014 [PHOTOS]
Julius Randle breaks leg during NBA debut