Natalie Sciver, Katherine Brunt, Sophie Eccleston and Sarah Glenn all claimed a pair of wickets each - with another dismissal for Mady Villiers - as the hosts were bowled out for just 96 at Westpac Stadium.
England - who won the toss and elected to field - then comfortably chased down their target, with Danni Wyatt top scoring with 33 as the tourists reached 99-3.
England wasted no time in drawing first blood as Sciver had New Zealand captain Sophie Devine caught for just two.
Sciver picked up her second almost immediately when Amy Satterthwaite was caught by Freya Davies for two.
Following that shaky start, New Zealand looked to have settled the ship as Hayley Jensen and Amelia Kerr began to build some momentum.
However that duo were eventually dismissed in quick succession, leaving the hosts floundering on 38 for four.
The wickets continued to fall at a fast pace, with those four England bowlers each claiming two scalps to restrict the home side to their modest total.
Tammy Beaumont and Wyatt shared an opening stand of 59 for England before Wyatt fell to Leigh Kasperek.
That platform proved enough for the visitors to easily chase down New Zealand's target off 16 overs, despite captain Heather Knight being run out for a duck.
The second match is scheduled for Friday.
Additional reporting by the Press Association.