New Zealand had set the Sri Lankans a formidable 405-run target but with sufficient time to get there on a non-threatening University Oval wicket.
The Sri Lanka batsmen, however, failed to show the necessary patience and were all out for 282 midway though the final day.
Resistance crumbled after Wagner, back in the side after a year out of Test cricket, captured the crucial scalp of Angelo Mathews, who had put on 56 with Dinesh Chandimal for the fourth wicket.
Sri Lanka struggle before rain intervenes
After being peppered by short deliveries targeting the rib cage, Mathews, on 25, was taken by surprise by a full-length delivery which rattled into his stumps.
His dismissal upset the rhythm of top-scorer Chandimal (58) who only lasted another 17 balls before he was beaten by Mitchell Santner.
"Neil Wagner really stepped up for us," captain Brendon McCullum said of the left-arm seamer playing his 18th Test.
"Once he got that breakthrough, got Angelo out, then we certainly relaxed the shoulders after that and were able to push home the advantage.
Gull-ible speed gun shows Wagner clock 160kph
"It was beautifully set up. Wags decided that at some stage he was going to bowl the miracle ball, try and hit the base of leg stump and in the end it split his defence."
Sri Lankan captain Mathews, who acknowledged being caught in a Wagner "trap", described the match as a learning experience for his young side.
"Most of the batters were solid in technique. It's just the mindset that we've got to shift around a little bit," he said.
"We've got so many inexperienced guys at the moment, it'll take a bit of time to get going."
Guptill ton gives New Zealand first day honours
The comprehensive victory gave New Zealand their fourth consecutive win over Sri Lanka in the past three years.
After a second-innings declaration at 267-3 by McCullum, Sri Lanka had more than five sessions to reach 405 or survive for the draw.
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.
New Zealand tried everything to break the partnership, including Trent Boult changing his grip which saw him lose control of one ball, forcing umpire Richard Kettleborough to duck for cover as it shot towards his head.
Finally it was Wagner who broke the partnership when he fired two short deliveries at Mathews.
The Sri Lankan skipper 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 soon after for 58 when trapped leg before wicket by Santner.
The 26-year-old batsman, in his 24th Test, decided to push his pad at the spinner and was hit plumb in front without playing a shot.
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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