Southee featured in a 61-run stand for the 10th wicket with Trent Boult as New Zealand advanced their overnight 286 for seven to be all out for 373.
The swing bowler then removed Kieran Powell without scoring in his first over as the West Indies reached lunch at 24 for one.
Kraigg Brathwaite was not out 14 with Shimron Hetmyer on 10.
After the quick removal of Powell, caught behind off an inside edge, Southee and Boult delivered four consecutive maidens, using swing and bounce to good effect.
The pair batted at better than a run a ball until Southee was out for 31, leaving Boult unbeaten on 37.
The West Indies made a promising start to the day when Kemar Roach removed Neil Wagner with the third ball of the morning, smartly caught one-handed by Shai Hope stretching to his left in the gully.
Tom Blundell, the debutant centurion in the first Test, advanced his overnight 12 to 28 before he played on to Shannon Gabriel leaving Southee and Boult to mount the final charge.
Boult had two sixes and five fours in his 37 off 27 deliveries.
Gabriel was the West Indies' most successful bowler with four for 119 while Roach took three for 58.
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