The visitors had been given a victory target of just 19 runs after they established a 92-run first innings lead earlier on Sunday before left-arm spinner Abdul Rehman was again the principle tormenter of New Zealand's batsmen, taking three wickets for 24 runs and creating enormous pressure.
Rehman, who was named man of the match, also combined magnificently with left-arm pace bowler Wahab Riaz, who ended with three for 38, while Umar Gul mopped up the tail to finish with three for 28.
Mohamam Hafeez and Taufeeq Umar then wrapped up the victory in just 3.4 overs to guide the visitors to 21-0 and a 1-0 lead in the two-match series. The second test begins in Wellington next Saturday.
New Zealand had gone to tea at 33 for no loss, however they crumbled after the break, losing seven wickets for 35 runs.
Tim McIntosh, who was dropped by Taufeeq Umar at first slip off Umar Gul and survived two confident lbw appeals before tea, was the first to go when he danced down the wicket to Rehman only to see the ball slide past his outside edge and be stumped by Adnan Akmal for three.
Brendon McCullum was next to go, though he had could be considered unlucky when he was given out by umpire Rod Tucker for 35 when he was caught by Akmal down the leg side off a Wahab delivery.
Television replays, however, showed he had not hit the ball with the delivery deflecting off his thigh pad.
Coincidentally McCullum was given not out in the first innings when replays suggested the ball had brushed his gloves.
The umpire review system is not being used in the series.
New Zealand then collapsed with Martin Guptill (11), Jesse Ryder (0), Ross Taylor (8) and Kane Williamson (1), all being dismissed for the addition for a total of one run, while captain Daniel Vettori was out lbw for three to reduce the hosts to 71 for seven. McIntosh was dismissed with the team score on 36.
Pakistan had been bowled out for 367 midway through the middle session, an overall lead of 92 runs after New Zealand were dismissed for 275 early on Saturday.
New Zealand had captured two wickets in two balls during the first session when Asad Shafiq (83) and Misbah-ul-haq (62), who had rescued the visitors on Saturday after they slumped to 107 for four, were both given out lbw with the score on 256.
The Pakistani tail, however, wagged with Akmal (44) and Rehman (28), who was dropped three times, combining for a 76-run partnership before they were both dismissed with the score on 332.
Gul and Tanvir Ahmed then frustrated the New Zealand attack with a 33-run partnership before Vettori trapped Gul in front for 17. Tanvir was the last man out for 18 when he was caught by Tim Southee off Chris Martin. Wahab was nought not out.
Brent Arnel finished with career-best figures of four for 95.
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