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.
Hales was the second wicket to fall when England were on 177 and just 47 runs short of victory.
England's domination began at the start of the day when Morgan won the toss and elected to bowl.
Midway through the 26th over, New Zealand were 93 for six while England were 192 for two at the same stage.
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.
Santner, who rescued New Zealand in the first match, hit a personal-best 67 while Nicholls was out for 55.
Woakes, who claimed Santner's wicket, was the most economical of the England bowlers taking three for 32 off his 10 overs.
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