Requiring 296 to win they knocked them off in 54.2 overs with seven wickets to spare.
To the delight of supporters who had been admitted to Headingley for free Jonny Bairstow completed the run chase with a four through the covers and a six over wide mid-on in successive balls of Michael Bracewell.
Bairstow smashed eight fours and three sixes to finish 71 not out off 44 balls, along the way making a 30-ball half-century, England's second fastest in Tests, one match after hitting England's second fastest century.
Joe Root made a more sedate but still swift unbeaten 86 from 125 balls.
It is England's first series victory since winning in Sri Lanka early in 2021 and followed a run in which they had won one and lost 11 of 17 Tests.
The transformation under new coach Brendon McCullum, and new captain Ben Stokes, sets England up nicely for the final Test of last year's uncompleted series against India. That begins on Friday (1 July) with England 2-1 down.
"It has gone really well," said Stokes. "To walk away with a 3-0 series win against the best team in the world is a pretty special start. It's a huge testament to the team.
"When I took over it was about more than results - it was about changing the mindset of the lads towards Test cricket, having fun, enjoying representing your country. Then the results will look after themselves.
"To have done it so quickly is unbelievable."
McCullum said: "I think it is a pretty simple game. Sometimes we can get caught up in the fear of failure and maybe try to protect something rather than actually enjoying the opportunity you've got to represent your country."
England, 183-2 overnight, needing 113 runs with eight wickets remaining, had to wait for victory as rain washed out the morning session. When play resumed Ollie Pope was soon bowled by Tim Southee for 82.
That brought in first innings centurion Bairstow to join Root, who had resumed on 55 not out. The Yorkshire pair, on their home ground in Leeds, wasted little further time knocking off the runs adding 111 in 87 balls.
That sealed a third successive winning run-chase in the series with England chasing down targets of 277, 299 and 296 with panache.
New Zealand captain Kane Williamson said: "It has been an incredible series. We certainly had our opportunities in all three matches but credit to the way the English team played.
"When they were under pressure they chose to counter-attack and they did that very well."
Former captain Root, who made 396 runs at 99 in the series, said: "It's such fun to play in and be part of. If you can't enjoy a series like this then what do you play for?"