that ended 37 balls short of their 50 overs, England overhauled the target in a rapid 37.3 overs.
The victory gives England a psychological boost going into the first Test on March 6 after earlier winning the Twenty20 series 2-1.
Following the early dismissal of Ian Bell for 24, Alastair Cook (46) and Jonathan Trott (38) made the most of the flat track and short boundaries at Eden Park to effortlessly carve out a 67-run second-wicket partnership.
After they fell in quick succession, Joe Root (not out 28) and Eoin Morgan (39 off 24) put on 56 for the fourth wicket before Morgan departed, caught by a leaping Kyle Mills on the boundary.
Jos Buttler came and went for three leaving Chris Woakes (not out three) to bat out the innings with Root whose undefeated stand lifted his average to an impressive 81.50 after eight ODI innings.
The match had belonged to England from the time Cook won the toss, elected to bowl and watched the New Zealand batting line up crumble.
They struggled against the pace and accuracy of Finn and James Anderson and were 67 for five after 25 overs while England at the same stage were untroubled at 120 for three.
Ian Bell set the early pace reaching 24 off 22 balls before he top-edged Andrew Ellis to Hamish Rutherford at deep square leg.
Tim Southee claimed Trott and Cook in the space of six balls in his second spell, both caught behind by BJ Watling who stood in as gloveman after Brendon McCullum suffered a sidestrain during his innings.
McCullum was again the standout performer with the bat for New Zealand, cracking 79 from 68 balls after his side had been firmly on the backfoot.
Finn at one stage had the remarkable figures of three for 10 off eight overs before McCullum distorted the statistics with two fours and a six in an over that cost the England quick 17 runs.
It was McCullum's third half century in as many matches, following innings of 69 not out and 74 in the first two matches, but he received little support.
Second-match centurion Ross Taylor made 28 after a 53-run partnership with Grant Elliott (24) for the fourth wicket and the only other batsman to reach double figures was James Franklin (13) who gave Graeme Swann his 100th ODI wicket.
Finn ended with the figures of three for 27 off nine overs, while Stuart Broad took two for 38 and Anderson one for 34.
England will play a four-day match against a New Zealand XI in Queenstown before the first Test.