It was New Zealand's only fifth series victory in Asia and first in 10 years, while Pakistan lost only their second Test series in the United Arab Emirates in the last decade.
Williamson, who resumed on the fourth day on 14, batted the entire day and grafted an unbeaten 139 off 282 balls with 13 fours.
Yasir Shah also equalled Pakistan's record of most wickets in a Test match by claiming 14 wickets in the second Test against New Zealand last week.
Openers Tom Latham (10) and Jeet Raval (0) yet again faltered as the visitors finished the day at 26-2 with New Zealand still trailing by 48 runs.
The veteran opener said that the ongoing third and final Test against New Zealand will be his last Test.
Boult provided New Zealand early success when the out-of-form Hafeez edged low to Tim Southee at second slip.
Yasir Shah equalled the Pakistan record for wickets in a Test with fast bowler Imran Khan, who achieved his feat in 1982 against Sri Lanka in Lahore.
Yasir became the first Pakistan player to take 10 wickets in a day as he took 8 in the first innings after seeing off New Zealand for 90 and two in the second after enforcing the follow-on.
Sohail's patient 147 off 421 balls spanned more than 9 1/2 hours and the maiden Test hundred by Azam (127 not out) defied New Zealand for two sessions after Pakistan resumed on its overnight 207/4.
The pair needed more than 60 overs to raise a 126-run third-wicket stand before Azhar's dismissal in the final session of Day 1.
Pakistan men's team were also heavily trolled by Twitterati after the humiliating loss against the Blackcaps.
The pressure immensely falls on Pakistan as they were at one point close to winning the match, but a sudden collapse saw the batsmen falling one after the other, only for them to lose the 1st Test.
Hasan's twin strikes in his fourth over of the day pegged back New Zealand's progress, while Shah wrecked the lower order after tea to dismiss the visitors for 249.
New Zealand recovered from Tom Latham's first-ball dismissal to reach 56-1 in its second innings at stumps, trailing by just 18 runs.
New Zealand lost wickets in a cluster after lunch when five batsmen were dismissed within 22 runs against Pakistan's relentless bowling attack.
Pakistan Cricket Board was trolled on Twitter after unveiling the trophy for the Test series between Pakistan and New Zealand.
Another uncapped player, batsman Saad Ali, was included after he scored 144 runs for Pakistan A against New Zealand A in the United Arab Emirates.
Imam-ul-Haq was rushed to a hospital for precautionary tests after he was hit on the grille by Lockie Ferguson's sharp bouncer.
"It’s not his job to give (the) action which was shown on TV," said an irate Sarfraz.
Danny Morrison and Shane Bond are the other New Zealand fast bowlers to perform hat tricks in ODIs.
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