Vettori ended Misbah-ul-Haq's resistance with the final delivery of his 40th over and then struck four balls before lunch to dislodge fast bowler Mohammad Aamer, leaving vice-captain Kamran Akmal to protect the tail while increasing Pakistan's lead from 173 in the middle session.
Either side of Vettori's double strike Southee removed the dangerous Umar Akmal for 77 to keep alive New Zealand's hopes of a fourth innings runchase later today.
Resuming on 347 for four, an overall lead of 99 runs, Pakistan advanced to 421 for seven at lunch.
Kamran Akmal will resume on 34 with new batsman Umar Gul.
New Zealand firmed as the only likely winners when Misbah's cautious seven from 65 balls ended via an uncharacteristic slog sweep against Vettori. He failed to connect, overbalanced and was smartly stumped by Brendon McCullum when the lead was 113 and after he had added just three to his overnight score from 43 balls.
The Akmal brothers were then charged with occupying the crease for the remainder of a 2-1/2 hour opening session.
Umar would have departed to the second ball of the day had McCullum or Ross Taylor been more alert.
Chris Martin induced an edge which flew between the wicketkeeper and Taylor, standing wider than usual at first slip, to the boundary to bring up his third half-century of the series.
Umar's milestone from 86 balls included five boundaries and the six which got him off the mark yesterday.
The slip-up behind the wicket was not too costly as Umar could not resist sparring at a delivery during Southee's first use of the new ball, and McCullum comfortably took the edge for his 15th dismissal of the series.
Aamer, who had batted cautiously for 52 balls, then slog swept Vettori to Martin in the deep.
Kept wicketless in Pakistan's first innings, Vettori had two for 70 from a marathon 48.3 overs while Southee had one for 64 from 29.
