After conceding an 18-run first innings lead, Boult grabbed three quick wickets to knock the top off the West Indies order before the pace attack ran through the rest of the line-up to dismiss them for 103.
Facing a target of 122 the New Zealand openers Peter Fulton and Hamish Rutherford negotiated the two overs before the close to reduce the deficit by six runs.
Boult, who could only manage one scalp in the first innings after bowling the home side to victory in the second Test, ended with 4-23 and Tim Southee wrapped up the tail to claim three wickets in an over.
Back-up seamers Neil Wagner and Corey Anderson shared the other three wickets to fall on a Seddon Park pitch that until then had offered more help to the spinners in the match.
Wagner claimed the prized scalp of first innings century-maker Shivnarine Chanderpaul who was brilliantly caught in the gully by Kane Williamson to reduce the visitors to 5-46 and then there was no stopping the home side.
The collapse came after a third century of the series to Ross Taylor could not stop New Zealand conceding a small first innings lead.
Taylor made 131, his 11th Test ton, as the home side reached 349 in reply to the visitors' 367.
Sunil Narine claimed career-best figures of 6-91 to take his Test wicket tally to 21, with 18 of those victims New Zealanders.
Taylor, who scored a double ton in the draw in Dunedin and hit 129 in the win at Wellington, has now made 493 runs in the series.
When he departed, Narine, who had the scalp of captain Brendon McCullum before lunch, went to work to claim the last three wickets to fall.
But any hopes the West Indies had of setting a challenging total were soon dashed by Boult.
He started the rot by bowling Kraigg Brathwaite between bat and pad and then Kieran Powell, who has endured a wretched series, was snapped up at third slip in the same over.
Kirk Edwards became his third wicket in 13 balls, but the key wicket was that of Chanderpaul, whose unbeaten 122 in the first innings was his 29th Test ton and had anchored their innings.
Wagner added his second before Boult returned to claim his fourth scalp as a dramatic session continued.
The fun and games were not finished with and Southee grabbed three wickets in an over, including his 100th Test victim, Veerasammy Permaul, to leave the home side on track for their first three-Test series win since 2006.