The tourists batted first, and after Jason Roy was dismissed for 42, Root (102) came in alongside Bairstow (138) as the pair formed a 190-run partnership - the second-highest against New Zealand in one-day history.
England started brightly with the ball, as Mark Wood and Chris Woakes took the wickets of Munro and Martin Guptill, both for ducks. Kane Williamson steadied the ship for New Zealand, before being caught from a Ben Stokes delivery.
Thomas Latham was next at the crease, and after being dropped by Moeen Ali, it was his century stand alongside Taylor that set New Zealand on their way to victory. Latham was dismissed before Colin de Grandhomme and Henry Nicholls ensured that the home side got over the line.
The deciding ODI will be played on Friday.