Offspinner William Somerville claimed two wickets and left-armer Ajaz Patel one to reduce Pakistan to 55-5.
Earlier, Kane Williamson was dismissed for 139 on the first ball of the fifth day before declaring the Black Caps’ second innings closed at 353-7 less than an hour later.
Fellow not out batsmen Henry Nicholls wasted no time in bringing up his third test century, getting there in the third over of the day. He finished not out on 126.
Nicholls was joined by Colin de Grandhomme when Williamson departed, and the all-rounder went in search of quick runs, making 26 off 19 balls before he was caught in the deep off legspinner Yasir Shah.
De Grandhomme and Nicholls added 62 for the sixth wicket in 6.3 overs together as the Black Caps’ lead grew from 198 to 260.
BJ Watling came and went for a golden duck, bowled by Shah, but couldn’t claim a hat-trick as Tim Southee saw him off.
Southee added 15 off 10 balls before the declaration came.
The Black Caps added 81 runs in nine overs of play, setting an aggressive tone.
A series win would be their first against opposition other than Bangladesh, the West Indies, or Zimbabwe since they beat England in 1999.
They have had two series wins in Bangladesh (in 2004 and 2008), two in the West Indies (in 2002 and 2014), and four in Zimbabwe (in 2000, 2005, 2011, and 2016) in that time.