Chris Woakes and Adil Rashid shared six wickets to restrict the hosts to an under-par 223 all out after Eoin Morgan won the toss in his 200th ODI.
Then Bairstow (104) hit six sixes and nine fours in his 58-ball century, the third-fastest by any Englishman, as his opening stand of 155 with Alex Hales (61) made memorably short work of New Zealand in this showdown decider which fell flat for most of a sell-out crowd at Hagley Oval.
England had 17.2 overs to spare as they extended a series-winning sequence which dates back almost 12 months exactly to a 3-0 success in the West Indies.
Bairstow was intent on completing the job in style, following his century in Dunedin three days ago with another brutal innings - including 22 off one over from leg-spinner Ish Sodhi.
Hales, back in place of the injured Jason Roy for his first match of the series and since he announced his white-ball-only future, was no slouch either and passed 2,000 ODI runs for his trouble.
Morgan’s men have won nine of their last 10 50-over series as they build towards a home World Cup next year.
Their latest, 3-2 against opponents who pushed them hard in a succession of tight finishes elsewhere, was never in any doubt from the moment New Zealand began fluffing their lines against some exemplary new-ball bowling from Woakes and Mark Wood.
Mitchell Santner (67) continued his prolific series, and Henry Nicholls (55) hit form too, in a seventh-wicket stand of 84 - but ultimately they managed only to delay the inevitable.
New Zealand lost Colin Munro in the first over to Woakes (3-32), for a second-ball duck in the first over when he mistimed an attempted leg-side flick to loop an easy catch for Jos Buttler.
Kane Williamson - returning to Yorkshire this summer - got an inside-edge on to his stumps when Wood went wide on the crease - and then England’s spinners took over, bowling in tandem and each straight through their 10 overs.
Rashid (3-42) had Tom Latham clipping to midwicket and foiled Martin Guptill’s attempt to anchor a faltering innings too when the opener poked a drive to cover three short of his 50.
Mark Chapman, back in place of injured lynchpin Ross Taylor, failed to score before trying to dab rather than defend a length ball from Moeen Ali - and paid with the loss of his off-bail.
With Colin de Grandhomme soon gone too, holing out off Rashid, at 93-6 Santner and Nicholls were the last hope of a telling partnership.
They delivered all that could reasonably be expected, putting on 84, Nicholls with a 73-ball half-century on his home ground and Santner capping his remarkable series from number eight with a new career-best.
Santner finished with 216 runs in four innings - at an average of 108 - before he fell to the first of two outstanding outfield catches in the 49th over off Woakes.
In a match which did not live up to its billing as a fitting tussle for a hard-fought series, Bairstow at least ensured a spectacle as he dominated the ruthless chase until he had to go after dislodging his own stumps as he made room to try to dispatch another boundary off Trent Boult.
It made precious little difference, despite the late departures of Hales and Morgan too before Ben Stokes hit a huge six off Sodhi for the winning runs.