Sri Lanka were 224-6 at the lunch break before Boult grabbed both Rangana Herath (six) and Milinda Sarawardana (29) in the first 20 minutes of the middle session before bowlers Mitchell Santner and Doug Bracewell completed the victory.
Santner bowled Dushmantha Chameera for 14 while Suranga Lakmal belted four boundaries in his 23 before he hit the ball straight back to Bracewell to end the match.
The hosts had earlier benefited from two brain explosions by Sri Lanka's top batsmen in the extended first session.
Both Angelo Mathews and Dinesh Chandimal looked comfortable on the benign pitch before Mathews failed to play a shot and was bowled for 25 by Neil Wagner.
Chandimal then also failed to offer a shot to a Santner delivery and was trapped lbw for 58.
The pair, who had put on 56 runs, were both dismissed with the score on 165.
Sri Lanka had resumed on 109-3 and needed to score 296 further runs from 98 overs to achieve the fourth-highest successful run chase in tests.
Chandimal brought up his 10th test half-century when he pulled a short ball from Doug Bracewell down to fine leg for a single, though he was not able to fully capitalise as both he and his captain fell shortly after the drinks break.
Kithuruwan Vithanage was also dismissed before lunch when he was trapped lbw to Tim Southee for a counter-attacking 38.
The second match of the two-test series starts at Seddon Park in Hamilton on Dec. 18.
(Reporting by Greg Stutchbury in Wellington; Editing by Ian Ransom)
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