After Nielsen beat Brian Elliott in the third round of the tiebreaker, Islanders netminder Nathan Lawson secured the victory with a save on Bobby Butler, handing New York its season-high fourth straight win.
Lawson stopped 28-of-31 shots to notch his first career NHL victory in his fifth start.
Butler and Ryan Shannon scored and Elliott turned aside 25-of-28 shots for the Senators.
NBC reportedly holds celebs hostage to Jimmy Fallon's show
Millions of Getty images now available for free via embed tool