Shortly after a faceoff win in the New York zone, Jared Cowen's blast from the slot was blocked by New York defenseman Ryan McDonagh, who sprawled to stop the shot.
But the rebound was gathered in by Neil, who flicked it past Henrik Lundqvist for the game-winner.
Nick Foligno and Erik Karlsson also scored for the eighth-seeded Senators, who will host Game 3 of the series Monday.
Craig Anderson earned the victory with 27 saves.
Anton Stralman and Brain Boyle found the net while Lundqvist turned away 29 shots for top-seeded New York.