Chasing a modest 135 for victory after pace bowler Haris Rauf choked out New Zealand with four wickets, Pakistan were in trouble at 69-4 but crossed the line in 18.4 overs for a fine start in their bid for a second championship.
Mohammad Rizwan top-scored with 33 while Shoaib Malik (26 not out) and Asif Ali (27 not out) guided them home.
The Group II contest was played against the backdrop of bad blood between the sides after New Zealand abruptly abandoned a limited-overs tour of Pakistan last month citing a security alert. England followed suit in a double blow.
But Pakistan, on a high after beating arch-rivals India in their Super 12 stage opener, channelled any lingering anger as they restricted New Zealand to 134-8 as Rauf picked up 4-22.
The contest was evenly poised until the 16th over with Pakistan at 98-5 but Tim Southee conceded 13 runs in the next as Asif smashed two sixes to turn the game on its head.
Malik struck a six and a four off Mitchell Santner to ensure Pakistan could complete the job in style.
Skipper Babar Azam kept faith with the eleven that hammered India and his decision to bowl after winning the toss paid off as they restricted New Zealand to 42-1 in the powerplay.
New Zealand struggled to release the handbrake as Pakistan continued to apply pressure and Rauf took two wickets in three balls in the 18th over to ensure there were no late fireworks.
"It's always good to win, we'll take this confidence forward in the tournament. The way Shaheen (Afridi) and Haris bowled was very impressive," Babar said.
"I think we gave them 10 runs too many, but it's cricket and that happens. We lost wickets early, but I want to give credit to Shoaib and Asif. Every game is crucial. We want to take it game-by-game and do well in all departments."
New Zealand opener Daryl Mitchell (27), skipper Kane Williamson (25) and Devon Conway (27) all got good starts but were unable to build on them.
"It's disappointing," Williamson said. "Unfortunately we couldn't nail things towards the back end but we were up against a very good side in Pakistan."
Pakistan meet Afghanistan on Friday while New Zealand face a potentially tricky test against a wounded India two days later.