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New Zealand, needing to make their biggest T20 total, 215, to win, reached 170 for nine in their 20 overs.
They battled to stay with the required run rate from early in the chase. England's bowlers were more demanding than New Zealand's had been, to back up a strong display by their batsmen in reaching 214 for seven.
England, who equalled the highest T20 international score made in New Zealand, peppered the crowd with 15 sixes, the third highest in T20 internationals, taking full advantage of Eden Park's joke boundaries at each end, where it is little more than a 60m blow to clear the rope.
Their tally was only exceeded by South Africa's 17 against England at Centurion four years ago and Australia's 16 against India in Barbados during the 2010 T20 World Cup.
Opener Alex Hales got things going with a booming blow to deep mid wicket in the second over from left armer Trent Boult, and Luke Wright, Eoin Morgan, Jonny Bairstow and Jos Buttler kept the heat on the New Zealand bowlers.
The hosts didn't help their cause with five dropped catches, two by returning former test captain Ross Taylor.
Wright plundered 42 off 20 balls. Four of the sixes were his.
With opener Michael Lumb, Wright shared a 60-run stand in only 26 balls.
Morgan and Bairstow - three and two sixes respectively - put on a rapid 81 off 43 balls for the fourth wicket before Buttler, the early-tour batting star after two flying half centuries against the New Zealand XI in Whangarei this week, belted 32 not out off 16 balls.
Boult, on debut, seamer Andrew Ellis and left arm spinner Ronnie Hira picked up two wickets each.
New Zealand debutant opener Hamish Rutherford took a couple of fours in Steven Finn's first over, and lifted captain Stuart Broad straight for six before departing next ball.
Brendon McCullum fell to a terrific running catch by Morgan at deep point and much rested on Martin Guptill and returning former captain Ross Taylor.
They put on 31 before Taylor slapped Finn to the longest boundary, at mid wicket, but didn't clear it, Bairstow taking the catch, and Taylor, who arrived to a standing ovation, was gone for 13.
At the halfway point, New Zealand were 88 for three, 21 behind England on comparative rate.
Guptill slapped medium pacer Wright to deep mid off on 44 and New Zealand's task gradually got steeper.
Broad picked up his best T20 figures, four for 24, while Finn took three and Wright followed his strong batting with two for 29 off his four overs.
The win was England's sixth in the countries' seventh T20 clash.
The second game of the series is in Hamilton on Tuesday.