England finished on 335 for nine after losing the toss and being put into bat as Bairstow blazed his way to 138 off 106 balls and Root grafted out a hard-fought 102.
The tourists' total should have been even bigger, but a dramatic late collapse put the brakes on their scoring as spinner Ish Sodhi led a stirring New Zealand fightback.
England were 267 for 1 in the 38th over when Bairstow's dismissal sparked the loss of six wickets for 21 runs.
Sodhi finished with four for 58, while Colin Munro and Trent Boult took two wickets apiece.
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Bairstow and fellow opener Jason Roy laid the foundation for England with a 77-run partnership before Roy departed on 41.
He then combined with Root for a 190-run partnership as New Zealand's pacemen struggled to find any movement.
England passed the 100 mark in the 16th over, with Bairstow smashing the ball onto the roof of the stands on his way to a 38-ball 50.
Bairstow amassed 14 fours and seven sixes before his innings finally ended after he tried to hit Munro out of the park.
It sparked a collapse as Jos Buttler, Eoin Morgan, Ben Stokes, Moeen Ali and Chris Woakes all departed without making double figures.
Root toiled at the other end as partners came and went, bringing up his 11th ODI century off 99 balls, with six fours and two sixes.
England lead the five-match series 2-1.
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