Nielsen's one-timer from the right circle on a power play got past Ondrej Pavelec and broke a 2-2 tie, putting the Islanders on the path to their fifth win in six games.
Rookie Anders Lee found the net in his NHL debut and Colin McDonald also scored in the win.
Kevin Poulin turned aside 23 shots for the Islanders.
Paul Postma scored twice and Pavelec made 34 saves for Southeast Division-leading Winnipeg.
Easer Egg Roll brings thousands to White House
Moore to attend retreat in to avoid Kutcher's wedding