Chasing 154, Jonny Bairstow (34) and captain Eoin Morgan (34*) also made noticeable contributions.
Earlier electing to bowl, the visitors never allowed the Black Caps to score freely and kept on taking wickets at regular interval to keep a check on the run-rate. Ross Taylor and (44) and Daryl Mitchell (30) did try to take the hosts to a formidable total but were kept in check by the England bowlers.
Chris Jordan was pick of the bowlers for Morgan as he gave away just 25 runs in his stipulated 4 overs and picked up crucial wickets of Colin Munro (21) and Tim Seifert (32). He was well supported by all the other four bowlers - Sam Curran (1/33), Tom Curran (0/25), Adil Rashid (1/31) and Patrick Brown (1/33).
The Three Lions started their chase cautiously as David Malan (11) and Bairstow played the ball on its merits and didn't try to go from the word go. Vince, who came after Malan's dismissal in the sixth over, however, played some beautiful shots and kept scoring boundaries (7) and sixes (2).
In the end, Morgan and Sam Bilings (14) shared an unbeaten 32-run stand and saw their team register a comfortable win.
For New Zealand, left-arm spinner Mitchell Santner was the lone wicket-taker as he picked three wickets, giving away just 23 runs in his four overs. However, he didn't get enough support from his teammates.
Vince, in the end, was adjudged the Man of the Match.
Both the teams will now lock horns in the second match at the Westpac Stadium, Wellington on Sunday.
Brief scores: England 154/3 (James Vince 59, Eoin Morgan 34*; Mitchell Santner 3/23) beat New Zealand 153/5 (Ross Taylor 44, Daryl Mitchell 30; Chris Jordan 2/25) by seven wickets.