Southee's wickets were his fifth five or more wicket haul in tests. Neil Wagner finished with three for 59 and Colin de Grandhomme one for 29.
Earlier, Pakistan had pulled themselves back into the test with a spirited batting display on the third morning. Quality contributions from Babar, Sarfraz Ahmed and Sohail Khan meant Pakistan had made good progress after starting the day at a parlous 76 for five in reply to New Zealand's 271.
The chief honours should go to the gifted Babar. He made three successive ODI centuries against the West Indies recently and has been earmarked as Pakistan's brightest batting prospect. Today it was easy to see why.
His strokeplay was often effortless and he avoided obvious traps, such as persistent short-pitched bowling to try and lure him into a false hook from Neil Wagner. By lunch he had batted 251 minutes, faced 181 balls and hit eight fours in a classy exhibition.
Babar shared a rousing early stand of 74 with wicketkeeper Sarfraz. Tim Southee's opening three-over spell went for 32 and Sarfraz was a happy pulled and cutter when the chance presented.
He was dropped, the ball going through Henry Nicholls hands above his head at mid wicket, and survived a mid wicket mixup, when sent back by Babar on 25.
Wagner eventually got Sarfraz spearing a catch to second slip.
Sohail used aggressive means, swinging consecutive Wagner short balls onto the bank at long leg and sharing a 67-run stand with Babar.
He became Southee's fourth victim, touching a catch to wicketkeeper BJ Watling at 37 and Wahab Riaz went shortly after, lbw to seamer Colin de Grandhomme, after his DRS review failed to save him.
New Zealand gave four overs to left arm spinner Mitchell Santner who, while tidy, showed it remains decidedly a seamers pitch.
