The 24-year-old Boult had taken 6-40, including five wickets in 15 balls, to help bowl the West Indies out for 193 in the first innings of the second Test.
He then took 4-40 in their second innings of 175 to put the hosts 1-0 up in the three-match series.
The left-armer, who scored 38 not out while batting at number 11 in New Zealand's 441, completed a remarkable match when he dived full length to his left to a full-blooded cut shot from Denesh Ramdin that sparked the second innings collapse after tea.
Shivnarine Chanderpaul was left on 31 not out after Tino Best's counter-attacking innings of unorthodox shots was ended by Boult on 21 runs, while Shannon Gabriel was bowled by the seamer for a duck to end the game.
The victory was the first for New Zealand since Brendon McCullum took over as captain late last year. They had drawn six and lost four of their previous 10 Tests before the victory inside three days at the Basin Reserve.
"I didn't envisage it happening the way it did today, but thoroughly pleased and proud of these guys," McCullum said.
"They've worked incredibly hard over a long period of time, and put ourselves in winnings positions in last few matches and haven't been able force the result.
"I thought we played some excellent Test cricket."
The win expunged the frustration they experienced in the drawn first Test at University Oval in Dunedin when they required 33 more runs to reach the 112 victory target only for rain to wash out the final session on the fifth day.
It also puts them in strong shape to win their first Test series against a major cricketing nation since 2006, when they beat West Indies 2-0 in a three Test series at home.
Since then they have only beaten Bangladesh or Zimbabwe in Test series.
New Zealand's pace bowlers had shared the workload on Friday as they captured 16 wickets in 67.4 overs, with Boult ripping through the West Indies batting order in their first innings.
"We didn't foresee 16 wickets falling in probably three hours, I think it was," West Indies captain Darren Sammy said.
"For a team that's been under the gun for some time, it could be very disappointing.
"I've guess we've found ourselves in these situations a few times and I believe we have the mettle in the dressing room to come back and keep remaining hungry because we value the fans back home."
Tim Southee captured three second innings wickets in quick succession after lunch to put the home side into a strong position to win the Test before Boult's catch lifted the team and allowed them to run through the tail.
"I don't know how that happened to be honest," Boult said with a smile when asked about the catch.
"I kind of threw my body up there and it literally went straight into my hands. Something pretty special and something I won't forget.
"It was brilliant if I can say that," he added while laughing.
The third match in the series begins at Seddon Park in Hamilton on December 19.