After New Zealand won the toss, Blair Tickner (4-64) and Jacob Duffy (3-56) spearheaded a fine display of swing bowling at Wellington Stadium to dismiss England for 222 in the 41st over.
The chase, however, was far from straightforward, especially after Devon Conway and Tom Latham both had the misfortune to be run out by deflections off the hands of the bowler.
Daryl Mitchell's runs had steered the hosts to their victories in Tauranga and Hamilton and when he departed for 44 to leave his team still 27 runs from their target on 8-196, the English sensed an unlikely victory.
Tickner, who made 18 not out, and Zak Foulkes, who finished unbeaten on 14, held their nerve to get the Black Caps across the line and secure New Zealand a 25th win in their last 27 completed ODIs on home soil.
Earlier, Duffy and Foulkes (2-27) bowled beautifully as Jamie Smith, Joe Root, Ben Duckett, Harry Brook and Jacob Bethell - all members of England's Ashes squad - were removed in little more than 10 overs to leave the visitors reeling on 5-44.
Jos Buttler and Sam Curran stopped the rot with a partnership of 53 before Tickner removed them both and it was left to two more bowlers in Brydon Carse, who made 36, and Jamie Overton to show the batters the way with another stand of 58.
In-form Overton scored 42 and 46 in the first two ODIs and was England's top-scorer for the second match in a row with his maiden ODI half-century, an aggressive 68 from 62 balls.
Tickner, who was named man of the match, took a four-wicket haul for a second consecutive match having been called in as a replacement when quicks Kyle Jamieson and Matt Henry were injured.
Overton broke New Zealand's 78-run opening stand when he diverted Rachin Ravindra's shot onto the stumps at the bowler's end with Conway (34) stranded out of his crease.
Ravindra was sent back for 46 in the following over to give Sam Curran his first wicket of the series and Will Young was caught behind off Overton before Carse repeated the deflection trick to run out Latham for 10.
Mitchell put on 34 with Michael Bracewell for the fifth wicket and 41 for the sixth with his captain Mitchell Santner, who showcased his power hitting by clubbing one six onto the stadium roof before holing out for 27.
England's bowlers kept chipping away, however, to set up a nervous finish for the home side with the smattering of fans around the arena cheering every run.
