England, having lost the toss, took to the crease and knocked up a score of 234, with Eoin Morgan's 48 the highest any of the players could muster.
Morgan - who passed 6,000 ODI runs during his time in the crease - hit two short of 50 to somewhat steady the ship, with New Zealand's best bowling performance coming from Ish Sodhi, who took three wickets.
New Zealand came into bat, carried through their innings by Kane Williamson, who ended unbeaten with his 11th ODI century from 133 balls.
Martin Guptill was dismissed for just four runs as England looked to exact revenge for their poor start, but Colin Munro - who could also have been taken by Root - rallied with seven fours and 49 runs to give his side momentum.
Williamson came in following Guptill's dismissal, and he was lucky to escape England's ruthless spin attack, which saw Mark Chapman (8), Thomas Latham (0), Henry Nicholls (0) and Colin de Grandhomme all dismissed in quick succession as Moeen Ali took three of the wickets while Adil Rashid chipped in with the fourth.
It was Chris Woakes next in the attack, and the pace bowler ran out Martin Santner, who had put New Zealand back on track with 41 from 54 balls.
The game came down to the final ball, when Williamson, who needed to hit a six, was unable to muster a response to Woakes's final delivery and the hosts finished on 230, granting England a series advantage for the first time.
The fourth ODI will take place on Tuesday evening.