Bairstow, whose batting heroics was key to ensuring England won the second test, picked up where he left off from Trent Bridge as he brought up his half century in 30 balls and then took England to victory on the last day with an unbeaten 71, finishing it off with a six.
A 111-run partnership with Joe Root, who was 86 not out, saw England reach the competitive victory target of 296 as they completed their first clean sweep of a home series in over a decade.
Rain delayed the start of play by 90 minutes but it then took just over an hour, and 15.2 overs, for England to canter to victory.
Resuming at 183-2 overnight, they needed 113 runs for victory and were very much in the driving seat even if Ollie Pope was dismissed with the fifth ball of the day.
Pope went for 82 as Tim Southee knocked over his off stump to offer New Zealand hope of a potential upset comeback but that was quickly dispelled by Bairstow's bashing of their bowlers.
Playing on his home ground, Bairstow hit eight fours and three sixes as he took to the bowling and left Root, who had been 55 overnight, little chance to get to a century.
It was the third successive test in which England reached a substantive target with an aggressive batting display. They won the first test at Lords as they chased down 277 runs while in the second test at Nottingham reached 299-5 in their second innings with Bairstow scoring 136.
"It was a pretty special start and to do it so quickly, phenomenal," said new England captain Ben Stokes.
The last time England swept a home series was a 4-0 success over India in 2011.
"We had an opportunity to win in all three tests but credit to the way England went about it. They chose to counter attack and did it very well," added New Zealand captain Kane Williamson.
England now move quickly onto completing last year's series against India after the fifth test at Old Trafford was cancelled because of Covid-19 cases. It will now be played at Edgbaston, starting on Friday.
New Zealand now prepare for limited overs matches against Ireland, Scotland and the Netherlands, although a number of the players are returning home.