The Black Caps, set 292 to win, finished on 297-2 with 26 balls remaining as they went 1-0 up in a four-match series.
Opening batsman Conway was 111 not out and Mitchell 118 not out, the pair sharing an unbroken third-wicket stand of 180 in New Zealand's first ODI against England since their agonising Super Over loss in the 2019 World Cup final at Lord's.
The teams will also face each other in the first match of the World Cup in Ahmedabad on October 5.
"We thought England's score was just above par, but the bowlers did well to restrict them on that surface," said Conway at the presentation ceremony.
"We just knew that if we built those partnerships it would get us over the line."
'Disappointing'
England captain Jos Buttler marked his 33rd birthday by top-scoring for his side with 72 in a total of 291-6.
Left-arm spinner Rachin Ravindra led a disciplined New Zealand attack with 3-48.
"Disappointing, at halfway I thought we actually had a pretty good score," said Buttler.
He added: "Credit to them (Conway and Mitchell), it was a fantastic partnership.
Looking ahead to the rest of the series, he said: "It's a quick turnaround so we'll give opportunities and look after guys as well.
"But we also want to play good cricket. We want to win games and we know we're up against a good team."
England had to wait until the 11th over for a breakthrough when leg-spinner Adil Rashid bowled Will Young with his first ball to leave New Zealand 61-1.
Rashid was briefly off the field for cramp and Mitchell greeted his return with a six and a four off successive deliveries.
Left-hander Conway completed his fourth ODI hundred, off 115 balls including 13 fours.
Mitchell then hammered Rashid for 16 off three balls - two sixes separated by a four - as he went into the 90s before going to an 84-ball century.
Conway ended the match in style with a straight six off Liam Livingstone as New Zealand recorded their third successive white-ball win over England this tour after ending a Twenty20 series level at 2-2.
Buttler fires England
Earlier, New Zealand held England to 291-6, with left-arm spinner Rachin Ravindra leading a disciplined attack with 3-48.
England captain Jos Buttler marked his 33rd birthday by top-scoring for his side with 72.
Four England batsmen passed fifty but none of them overwhelmed New Zealand's bowlers.
Together with the returning Ben Stokes (52), Buttler shared a fourth-wicket stand of 88 after England lost the toss.
Livingstone's typically dashing 52 off 40 balls added late impetus after opener Dawid Malan made 54.
Harry Brook was given another chance to push his World Cup claims after being drafted in as opener alongside Dawid Malan following injuries to Jason Roy and Jonny Bairstow.
He was left out of the provisional squad for next month's title defence in India.
But it was Malan who dominated an opening stand of 80 before being bowled off his pads for 54 by Ravindra.
Brook fell for a subdued 25 in the next over when he gloved a Lockie Ferguson bouncer to New Zealand captain and wicketkeeper Tom Latham.
Buttler and Stokes consolidated, with England's limited-overs skipper completing a 48-ball fifty.
Stokes, in his first ODI since reversing his retirement from the format ahead of the World Cup, went to the landmark with a six over midwicket off Ravindra.
But the bowler had his revenge next ball, when Stokes slapped him to cover.
Livingstone holed out off Tim Southee's slower ball and the veteran seamer also dismissed Buttler.
The series continues at Southampton on Sunday.
