Bairstow's 104 off 60 deliveries, backed by 61 from fellow opener Alex Hales transformed the winner-takes-all showdown from an expected tight contest into a one-way event.
New Zealand, sent in to bat, struggled to 223 with the last wicket falling on the penultimate ball and England knocked off their target in 32.4 overs with Ben Stokes finishing the match with a six.
Both sides were hit with late replacements but with marked differences.
England lost Jason Roy to a back spasm and his replacement Hales opened with Bairstow in a 155-run stand for the first wicket.
Hales was the second wicket to fall when England were on 177 and just 47 runs short of victory.
Eoin Morgan had a short stay for eight while Ben Stokes was 26 not out at the end with Joe Root on 23.
England's domination began at the start of the day when Morgan won the toss and elected to bowl.
Bairstow toyed with the best of New Zealand's bowling stocks as he smacked nine fours and six sixes.
He eventually brought about his own downfall by belting the stumps as he stepped back to cut a wide Trent Boult delivery.
Hales was at the crease for 74 deliveries for his 61 which included nine boundaries.
After the first four ODIs, which had the series level at 2-2, the England bowling attack had the New Zealand batsmen well worked out.
Mark Wood bowled Kane Williamson for 14 before spinners Adil Rashid and Moeen Ali stepped up to decimate the middle of the New Zealand innings.
Rashid, who finished with three for 42, accounted for Tom Latham (10), Martin Guptill (47) and Colin de Grandhomme (six).
Mitchell Santner and Henry Nicholls put on 84 for the seventh wicket to give the innings some substance.
Santner, who rescued New Zealand in the first match, hit a personal-best 67 while Nicholls was out for 55.
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