Chapman posted his third one-day international century and rebuilt the innings with Mitchell after the hosts had been three wickets down in the 13th over of Saturday's match, struggling against bounce and swing from the Pakistan pace bowlers.
Batting at No.6, Muhammad Abbas, the first Pakistan-born player to play for New Zealand, made 50 from 24 balls - the fastest half century by a player on debut in one-day internationals.
Babar Azam led Pakistan's run chase with 78 from 83 balls but when he was out Pakistan slumped, losing seven wickets for 22 runs.
For much of their innings Pakistan were ahead of New Zealand, 0-64 after 10 overs where New Zealand had been 2-34.
Usman Khan and Abdullah Shafique put on 83 for the first wicket, Babar and Mohammad Rizwan added 76 for the third wicket then Babar and Salman Ali Agha contributed a further 85 for the fourth wicket.
Pakistan were 3-249 in the 39th over and still well on target.
New Zealand captain Michael Bracewell was forced to bring back his strike bowlers Will O'Rourke and Jacob Duffy. O'Rourke offered a short ball to Babar which he hooked to Mitchell on the boundary.
Duffy then ran out Tayyab Tahir (1) and bowled Irfan Khan first ball as Pakistan slumped to 6-253 in the 40th over. Pakistan's hopes faded as wickets dwindled and the run-rate rose. Pakistan were 9-271 when Salman was out for 58 from 48 balls.
"We started the second innings with good intent and we got the things you need for a big chase," Rizwan said. "But at the end the pressure was very big because as you get near to the end, the pressure becomes more."
Pakistan won the toss, bowled and were on top early through the efforts of seam bowlers Naseem Shah, Akif Javed, Mohammad Ali and Haris Rauf.
New Zealand slumped to 3-34 in the 13th over, bringing Chapman and Mitchell together. Runs were hard to come by at first but after bowling his seamers through the first 25 overs, Rizwan was forced to go to his fifth bowler. The New Zealanders cashed in.
Chapman and Mitchell took 67 runs from the five overs bowled by Salman and 51 runs from five overs bowled by Irfan Khan.
Irfan had his revenge dismissing both Chapman and Mitchell. Chapman's 132 came from 111 balls and Mitchell's 76 from 84 deliveries.
"It was certainly pretty tricky to bat early on," Chapman said. "We were probably aiming for a total lower than we got in the end. But it's something we pride ourselves on, being able to adapt to the conditions then putting up a score and just scrapping."
Pakistan lost the preceding five-match T20 series 4-1 by margins of nine wickets, five wickets, 115 runs and eight wickets.
