Chapman and Dean Foxcroft (31) shared a century stand for the third wicket and New Zealand reached 3-179 to win in 18.2 overs, only a day after the visitors were bundled out for 90 runs to lose the second game by seven wickets.
An inexperienced Black Caps bowling attack, that featured debutant fast bowlers Will O'Rourke and Zak Foulkes, had earlier pinned down Pakistan to 4-178 after Michael Bracewell won the toss and chose to field first.
It was an impressive performance by New Zealand who showed their depth after nine of their frontline players were unavailable because of the Indian Premier League.
Pakistan's batting mainstays -- captain Babar Azam (37) and Mohammad Rizwan (22) -- couldn't convert good starts with Rizwan forced to retire hurt (right hamstring) midway through the innings.
Pakistan felt Rizwan's absence in the field too when makeshift wicketkeeper Usman Khan dropped one of the three catching chances off Chapman, who smashed nine fours and four sixes.
"Maybe we were 10 runs short," Babar said. "We started well with the ball in the first six overs, but when you drop a set batter's catch then he will take away the game from you. Chapman has always done well against us."
New Zealand lost both their openers within the powerplay as Tim Siefert was bowled by Abbas Afridi (2-27) off a delivery that kept low and Naseem Shah got Tim Robinson's middle stump uprooted with a brutal yorker.
But Chapman then took control against Pakistan spinners Abrar Ahmed (0-31) and Shadab Khan (0-24) before he cut loose against Naseem (1-44) and Shaheen Shah Afridi (0-37) in the death overs.
He smashed Naseem for two sixes and two fours in one over to raise the century stand in the 16th over before Foxcroft top edged a pull against Abbas in his return spell with the Black Caps needing only nine for victory.
Earlier, New Zealand were also scrappy in the field when Seifert dropped Saim Ayub (32) down the legside in the second over as the left-hander provided Pakistan with a brisk start of 0-54 in the power play.
Ayub also successfully overturned an lbw decision against him before leg-spinner Sodhi (2-25) stemmed the flow of runs soon after the power play when he had him caught in the outfield.
Bracewell, who bowled two overs of off spin in the power play, returned and took the prized wicket of Babar when the Pakistan skipper offered a tame catch at backward point.
Shadab Khan's brisk 41 off 20 balls lifted Pakistan's total that included 19 runs in Bracewell's last over of his spell.
