Chasing 224 to win, opening batsmen Jonny Bairstow and Alex Hales – a late replacement for Jason Roy who suffered a back spasm – took them to 155 as the doddle unfolded.
Bairstow was a tour de force, reaching his century from 58 balls - England's third fastest in ODIs.
At one stage he pumped five sixes from eight balls off Ish Sodhi who finished with one for 78 from 7.4 overs.
Bairstow eventually reached 104 from 60 balls, his fourth ton in the format and second in succession. Earlier he inspired in the field with a one-handed catch of Tim Southee running around the deep mid-wicket boundary.
Hales anchored the other end, compiling 61 off 74 balls as the pair milked the bowling and capitalised on a New Zealand side failing to ignite.
The hosts' unbeaten record in eight ODIs at Hagley Oval ceased.
The platform for the triumph was laid early when England's bowlers constricted New Zealand's top order.
Any scope for momentum was siphoned off by the 27th over when New Zealand concertinaed to 93 for six. With Colin de Grandhomme's exit, New Zealand had lost four wickets for 33 runs in 60 balls.
Mitchell Santner came to the rescue again in his finest ODI batting series, but his acceleration was tethered to a lack of wickets-in-hand.
The bowling all-rounder compiled 67 from 71 balls, his highest score in the format, eclipsing the 63 not out in the loss at Mt Maunganui.
His 84-run seventh-wicket partnership with Henry Nicholls (55 from 81 balls) offered spine across 17.2 overs.
New Zealand were all out with a ball to spare.
England chose to field and, after the first 10-over powerplay left New Zealand struggling at 27 for two, captain Eoin Morgan persevered with the spin of Adil Rashid and Moeen Ali.
Both had bowled out after 34 overs. Rashid finished with three for 42 and Ali one for 39.
Hopes were high New Zealand's heroics in Dunedin, led by Ross Taylor's 181 not out, could be repeated in front of 9012 fans.
Such optimism took an early hit when the quadricep injury Taylor suffered during the fourth match ruled him out. The prospect of running extra twos and threes on a relatively bigger ground would have been daunting.
England presented tight line-and-length bowling and committed fielding. It was straitjacket time for New Zealand's batsmen with no room to free their arms and attack width.
The Black Falcons from the Royal New Zealand Airforce aerobatics team made a cameo at 400 feet over Hagley Oval to mark the start. It was touch-and-go between them and Colin Munro as to who disappeared fastest.
Munro exited for a duck from the third ball of the game, courtesy of a leading edge from the bowling of Chris Woakes. The ball found a resting place in Jos Buttler's gloves.
Woakes deserved particular credit for his opening spells across this series - 5-0-14-2, 5-1-18-2, 5-1-13-1, 5-2-14-1, 6-1-13-1 – such consistency is an asset any team would welcome.
Martin Guptill anchored the innings with 47 from 68 balls early. Kane Williamson (14 from 25 balls) and Tom Latham (10 from 19 balls) tried to offer support but were no match for England's pressure.
Mark Chapman, batting at No.5 in Taylor's absence, looked ill at ease against the spin of Ali coming around the wicket. He played inside the line to such an extent that Ali suddenly looked like Muttiah Muralitharan.
England hunted the same target to win by six wickets with 73 balls to spare at Mt Maunganui in the second match of the series.
