The tourists added 53 runs in the morning session before being bowled out for 690 -- bettering New Zealand's previous highest score of 680 made against India in February -- to take a 339-run lead on first innings.
Boult then removed batsman Shan Masood, Azhar Ali and Younis Khan in an hour of lively swing bowling on the fourth day.
Earlier, resuming on 637-8, New Zealand frustrated Pakistan further with No. 8 batsman Mark Craig scoring 65 off 85 balls before he was caught and bowled by Mohammad Hafeez.
Ish Sodhi (22) also hit a six and a four before he looped a catch to Khan in the slips off a mistimed sweep to give legspinner Yasir Shah (4-193) his fourth wicket of the innings.
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Rahat Ali was the pick of the bowlers with 4-99 while Shah conceded most runs for his four wickets. Fast bowler Mohammad Talha and left-arm spinner Zulfiqar Babar had forgetful test match with both conceding 271 runs in between them without taking a wicket.
Boult then dealt a severe blow just before lunch when Khan was out lbw for a golden duck for the first time after eight years before Misbah-ul-Haq survived the hat-trick ball.