Even Brendon McCullum's third successive half-century was only able to drag the score to 185 all out, albeit on a pitch which had offered plenty to well-directed seam bowling.
England's two premier exponents Finn and James Anderson soon reduced the hosts to a hapless 11 for three after Alastair Cook had won the toss.
Finn's first of three wickets for 27 was his 50th in ODI cricket, and for good measure Graeme Swann was later to take his 100th - only the eighth Englishman to do so in this format.
Captain McCullum has been England's one stumbling block throughout these limited-overs matches, and he was at it again today with six fours and five sixes in his 68-ball 79. His lone hand was forced, though, and he was last out in only the 44th over when he tried to pull another maximum off Swann only to be well held just inside the deep-midwicket boundary by Anderson.
Finn began England's dominance by beating BJ Watling twice in three balls in his first over, with bounce and away movement. With a third similarly testing delivery, he got his first reward - taking the shoulder of a defensive bat for a neat catch by Swann at second slip. Then after a Finn maiden to Hamish Rutherford, Anderson struck with a length ball which held its line and had Kane Williamson edging behind on the back-foot defence.
Rutherford, in particular, was enduring an unequal struggle. He managed just two runs from 19 balls, eventually falling to Finn. The Middlesex man allowed him a little width for the first time but, rather than being punished, had the left-handed opener edging to the wicketkeeper as he tried to seize on a rare scoring opportunity.
Ross Taylor and Grant Elliott dug in to attempt a recovery. But after a hard-working stand of 53 in 15 overs, their industry was squandered via a mix-up over a second run which left Elliott sent back and stranded mid-pitch. When Taylor then became the third caught-behind departure of the innings - umpire Chris Gaffaney's decision upheld after a DRS process which was unable to disprove it - it was hard to envisage how New Zealand could reach a competitive total.
McCullum would still have his say, though, and in the 32nd over hit the the first six of the innings - down the wicket to Swann, over long-on. Another feasible partner came and went, James Franklin poking a catch back to a diving Swann off the back foot to bring up the off-spinner's ODI century; then McCullum's brother Nathan edged the returning Finn to slip at the start of the batting powerplay. Finn had figures of 8-3-10-3 at that stage.
McCullum punctured his pride by clubbing him for two pulled fours and a six over long-off in successive balls in a ninth over which cost 17, and New Zealand's lynchpin continued his one-man counter-attack too. But he needed lasting support, and it was never forthcoming.