Earlier, Alex Hales had announced England's intent after being put into bat by smashing their first six high over deep mid-wicket off Trent Boult in the second over.
Hales was stumped off Ronnie Hira's left-arm spin, but Wright upped the pace and a second over from Hira cost 21 runs.
Michael Lumb had scored only three, from just six balls faced, in England's first 50 runs. But he joined in with a swept four and straight six off Hira.
Wright was then caught in the leg-side deep off Andrew Ellis and then Lumb miscued an attempted hook at Mitchell McClenaghan to short fine-leg.
New Zealand were not helping themselves in the field, though, as Hira dropped Bairstow on 22, but the most obvious chance fell to McClenaghan who appeared not to sight one properly at short third-man when he put down Morgan on 33.
The combined cost for the Kiwis' third and fourth drops was 29 runs, Morgan miscuing Hira into the off-side to give Taylor an unmissable opportunity and Bairstow unable to clear Guptill at long-on off Boult.
Jos Buttler nonetheless ensured England surged past 200 and beyond with a stunning unbeaten knock of 32 off only 16 balls.
