Bairstow’s 104 off 60 deliveries, along with 61 from Alex Hales saw England swiftly meet the target of 224 set by the Kiwis.
England had 104 balls to spare in the emphatic series-deciding win.
The tourists eclipsed New Zealand. In the middle of the 26th over in each innings when New Zealand were 93 for six while England were 192 for two at the same stage.
"Our bowlers really did set the tone," said captain Eoin Morgan. After winning the toss he opted to make New Zealand bat first.
Right arm fast bowler Chris Woakes was named man of the series after setting the pace for England’s bowling unit with three for 32 off his 10 overs while spinner Adil Rashid finished with three for 42.
England were helped by the absence of New Zealand batsman Ross Taylor, who was ruled out with a thigh injury that he aggravated when he scored an unbeaten 181 in the five-wicket win in the fourth game on Wednesday.
New Zealand’s Martin Guptill was the only batsman in their top five to provide any resistance with 47, before Nicholls and Santner ensured New Zealand had a total to bowl.
But from the outset their 224 never looked as if it would be enough against the powerful England batting lineup.