At stumps on Tuesday at Sabina Park, New Zealand were 260 runs ahead with eight second-innings wickets in hand, having bowled their hosts out for 262.
Southee and Craig, making his Test debut, were the main aggressors, picking up four wickets apiece.
The Windies had a small win late in the day, restricing NZ to 14-2 after scalping Peter Fulton for a duck – on the second ball of the inning – and Kane Williamson (two).
Resuming on 19 without loss at the start of the day in reply to New Zealand’s 508-7 declared, the West Indies made a watchful start, taking the score along to 60 before Craig claimed two wickets in three balls.
The off-spinner trapped Kieran Powell lbw for 28, and then had Kirk Edwards caught at first slip by Ross Taylor for a second-ball duck.
There was further delight for the tourists in the next over, as Sodhi held a return catch to dismiss Darren Bravo, who departed without scoring to leave the Windies toiling at 61-3.
That brought Shivnarine Chanderpaul to the crease, and the famously stubborn left-hander proved his worth to the side yet again, staging a recovery with Chris Gayle.
Gayle, playing in his 100th Test, took his tally to 64 before Southee struck a crucial double blow, catching the power-hitter’s outside edge before trapping Marlon Samuels lbw two balls later.
Denesh Ramdin (39), captaining the side for the first time, added 72 with Chanderpaul before flicking Southee’s slower ball down the leg side into the grateful gloves of BJ Watling.
Craig accounted for Kemar Roach and Sulieman Benn, either side of Boult removing Jerome Taylor, to finish with 4-91.
Southee wrapped up the innings by having Shane Shillingford caught behind to end with 4-19 from 16.2 overs, Chanderpaul once again left stranded, unbeaten on 84.
The Windies picked up some much-needed momentum late, by ensuring opener Fulton failed for the second time in the Test match, before ousting first-innings century maker Williamson cheaply.
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