Left-arm seamer Boult bowled Chris Jordan and then Liam Plunkett on the final two deliveries of the 18th over to reduce the visitors to 168 for eight in their pursuit of 197 for victory.
Adil Rashid kept out the hat-trick delivery on the first ball of the final over but there was too much for the legspinner and Chris Wood to do and they finished on 184 for nine.
“It was a must-win for us in terms of the series,” New Zealand captain Kane Williamson said in reference to their seven-wicket loss to Australia in their opening match.
“This is a very strong England side and we had to play some really good cricket.”
Australia have already qualified for the final at Eden Park on Feb. 21 after winning all three of their matches so far.
Williamson anchored New Zealand´s 196 for five with 72 while he also shared in an 82-run second-wicket partnership with Martin Guptill, who scored 65 from 40 balls.
Alex Hales got England´s run chase off to a fast start with 47 from 24 balls before Dawid Malan gave England hope for victory as he scored 59 from 40 deliveries.
Malan´s dismissal in the 17th over with the visitors needing 39 runs for victory, effectively ended their chances.
“It was maybe 10 to 15 runs too many .. and we didn´t bowl as well as we could have,” England captain Jos Buttler said.
“We got a fantastic start with Alex and that partnership with Malan, we needed to just extend a bit longer then we didn´t get those cameos in the middle order.”
New Zealand face in Australia the next game of the series in Auckland on Friday.
Mark Chapman, who has also played international cricket for Hong Kong, and debutant Tim Seifert then did what they had been brought into the side to do with quick-fire cameos in the middle order.
Chapman scored 20, while wicketkeeper Seifert hit two sixes in his six-ball innings of 14 to take the hosts to the equal highest Twenty20 international score on the ground.
“We were able to build partnerships throughout that gave us momentum and the two debutants were outstanding,” Williamson added.
“They showed their power which allowed us to get an above-par total.”