The Blacks Caps looked in deep water at 73-4 in reply to Pakistan's 193-5 but Chapman and all-rounder Jimmy Neesham, who scored 45 not out, combined to get the tourists over the line with four balls to spare and no further losses.
It was New Zealand's second highest successful run chase in T20s and all the more notable for being achieved by a weakened team led by a stand-in coach against a bowling attack that fired Pakistan to the final of last year's World Cup.
"I think we were a little bit under the pump there after the first 10 overs of our innings," Neesham told reporters in Rawalpindi.
"Myself and Chappy decided we'd either go down fighting and get bowled out or give it a good crack. I think Chappy batted outstandingly, he's been hitting the ball beautifully over the last two or three weeks.
"To walk away from a series two-all against a very, very good team at home is pretty pleasing."
See the scorecard of the final T20 at Flashscore
Pakistan looked to be in a strong position to clinch the series after opener Mohammad Rizwan's unbeaten 98 led the hosts to a decent tally.
Lefthander Chapman, however, has enjoyed a remarkable tour, dismissed only once in five innings and racking up 290 runs -- a record tally for a five-match T20 series.
As well as taking player of the match and player of the series honours, his performances with the bat earned him an immediate call-up to the squad for the one-day international series that starts in Rawalpindi on Thursday.
"The way Mark's played against one of the best T20 bowling attacks in the world has been nothing short of outstanding," said coach Gary Stead, who will rejoin the squad for the one-dayers after skipping the T20s.
"His match winning knock in the fifth and deciding match was particularly special and we're happy to have someone in such strong form added to our ODI squad."
"I had to check back into my room," Chapman said. "I had already had paid my bill and checked out.
"I got back here, had to get another room key ... Same room, thankfully. So, it feels like home."
