West Indies, set a target of 308 at Kensington Oval when New Zealand declared overnight on 331-7, was bowled out for 254 with 13.4 overs remaining yesterday. Southee took 3-28 and Boult 3-48.
Off-spinner Mark Craig was also impressive with 3-84. Neil Wagner contributed 1-50.
All-rounder Jason Holder top-scored for the West Indies with 52. Darren Bravo (40), Shane Shillingford (30 not out), captain Denesh Ramdin (29) and Shivnarine Chanderpaul (25) also got starts without carrying on.
"This series is groundbreaking for this group. We're very proud of what we've been able to achieve and I can think of worse places than Barbados to be able to celebrate it."
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McCullum's declaration set the hosts an enticing target of 308 in a possible 98 overs and his new ball pair of Boult and Southee gave him the best possible start.
Boult struck in his second over when Kraigg Brathwaite (6) played no stroke to a big inswinger and lost his off stump after the ball kissed his pad to make it 7-1.
Southee claimed the prize wicket of Chris Gayle soon afterward as the powerful left-hander was bowled off pad and bat as he aimed an expansive drive to leave the hosts in serious trouble at 31-3.