New Zealand had set Sri Lanka a formidable 405-run target to win but with sufficient time to get there on a non-threatening University Oval wicket.
But Sri Lanka's batsmen failed to show the necessary patience as they were all out for for 282 midway through the last day.
Sri Lanka's two top batsmen, Dinesh Chandimal and Angelo Mathews, showed some resistance at the start of the day putting on 56 for the fourth wicket before both fell in quick succession.
The comprehensive victory gave New Zealand their fourth consecutive win over Sri Lanka in the past three years.
After a bold second-innings declaration at 267-3 by Brendon McCullum, Sri Lanka had more than five sessions to reach 405.
When they resumed Monday at 109-3, and with an extra half-hour tacked on to make up for three rain breaks the previous day, Sri Lanka were 296 runs behind.
But any hopes of a salvage job were built around the experienced Chandimal and Mathews digging in for a long stay at the crease.
It was Wagner, recalled after a year out of Test cricket, who broke the partnership when he fired two short deliveries at Mathews.
The Sri Lankan captain shaped up as if expecting a third similar delivery only for Wagner to bowl full and the ball speared off the left pad and on to middle stump.
Chandimal was gone less than three overs later for 58 when trapped leg before wicket by spinner Santner.
Kithuruwan Vithanage decided attack was the best policy, batting at a run-a-ball pace to take the target below 200.
With six fours and a six to his name he reached 38 before Tim Southee had him lbw.
At lunch, Sri Lanka were 224-6 with the final four wickets falling for the addition of 58 runs in 11 overs after the interval.
For New Zealand, Southee took three for 52 while Trent Boult, Wagner and Santner finished with two wickets each.
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