"We knew it would be a tough chase. We knew we had to get a little bit out of the surface, and the two seamers set the tone for us."
Pakistan won the first match by six wickets while the third and final match is on Friday, also in Karachi.
Pakistan lost openers Fakhar Zaman (nought) and Imam-ul-Haq (six) by the fourth over. Azam and Mohammad Rizwan (28) added 55 for the third wicket but it was not enough.
Agha Salman scored 25 before he was run out in a mix-up with Azam while none of Mohammad Nawaz, Usama Mir or Mohammad Wasim stuck around for long.
Azam hit eight fours and a six in an innings spanning 114 balls before he was finally stumped off spinner Ish Sodhi in the 43rd over.
"We came back well with the bat," said Azam. "But when the spinners came on, it was turning sharply. Credit to them."
Tim Southee claimed 2-33 for New Zealand while Sodhi took 2-38.
Earlier, left-arm spinner Nawaz picked up four wickets to slow New Zealand after the visitors won the toss and batted first.
Nawaz finished with 4-38 as New Zealand slumped from a strong position at 183-1 to 206-6, losing five wickets for 23 runs in the space of 40 deliveries.
New Zealand were lifted by a second-wicket stand of 181 between Conway and Williamson before fast bowler Naseem Shah (3-58) broke the stand when he dismissed Conway off the last ball of the 30th over.
Conway hit 13 fours and a six off 92 balls in his second ODI hundred.
Nawaz then took three wickets, dismissing both Daryl Mitchell and Tom Latham cheaply in one over before clean bowling Williamson in his next.
Williamson, dropped on 53 and 54 off Wasim, struck 10 boundaries in his 100-ball knock.
Mitchell Santner, one of just three New Zealand batters to reach double figures, was the last man out for 37.