India lost their top three batsmen in the post-lunch session after New Zealand skipper Brendon McCullum became the first Kiwi batsman to hit a Test triple-hundred.
Riding on McCullum's historic 302 and debutant James Neesham's unbeaten 137, the hosts declared their second innings at 680 for eight in 210 overs with an overall lead of 434 runs at the Basin Reserve.
Starting at 10 for no loss post-lunch, India had the simple task of batting out two sessions to save the Test. But as has been the story in this tour so far wherein they haven't managed to win a single game, the visitors botched it up.
Southee added another to his wicket-tally, breaking the 44-run third-wicket partnership between Cheteshwar Pujara (17) and Virat Kohli.
A sharp bouncer kissed Pujara's glove on its way to wicket-keeper BJ Watling, bringing an end to a disappointing tour for him.