Sunday's victory keeps alive the visitors' hopes of drawing the four-match series.
New Zealand, who lost the first two matches heavily, opted to bat and had Finn Allen (83) and Glenn Phillips (69) to thank for helping them past the 200 mark.
The pair combined for an 88-run partnership off 47 balls in the Black Caps' 5 for 202.
With a sizeable target to chase, England lost their opening four batters cheaply, before captain Jos Buttler scored a battling 40 and Moeen Ali contributed 26 runs to give them hope.
However, once Buttler departed after being caught-and-bowled by Mitchell Santner, England wickets continued to fall at a steady pace as the hosts were bundled out for 128 with nine balls left.
"Finn showed his class today and Glenn has been outstanding for us for a long time, especially on tricky surfaces. He made it look like a completely different surface to everyone else," New Zealand captain Tim Southee told the BBC.
"So we got a decent score and then with the ball we were more clinical and also had the advantage of having 20 overs watching England bowl.
"It is nice to keep the four-match series alive and in another couple of days we will be back into it."
Buttler praised fast bowler Gus Atkinson, who added two more wickets to the four-wicket haul he claimed on his debut in the second T20I.
"Credit to New Zealand, they outplayed us. It was a good toss to win, we let them get too many. We couldn't break that Allen-Phillips partnership and they just took the game away," Buttler told Sky Sports.
"I'm delighted for Gus. It was great for him to get exposure out there, he looked very comfortable."
The fourth and final match will be played at Trent Bridge on Tuesday.