Jason Roy and Jonny Bairstow added their third consecutive 100-run stand for the World Cup.
Jonny Bairstow brings up his 9th ODI hundred and second in the World Cup.
Trent Boult started New Zealand's fight back by picking Joe Root and Jos Buttler in his second spell.
James Neesham was the pick of the New Zealand bowlers, finishing with 2 for 41 in his 10 overs.
Chris Woakes provided the perfect start for England by trapping Mark Nicholls LBW in the very first over.
Mark Wood got a fingertip on a Ross Taylor drive and the ball deflected onto the stumps at the bowler's end to catch Kane Williamson short of the crease
New Zealand lost their two premier batsmen - Williamson and Root - through run-outs.
Tom Lathan returned to form with a fifty and was the only one to show any kind of resistance.