Starting out on 293-5 overnight in their second innings, and still needing another 10 runs to make the Kiwis bat again, the tourists lasted another 29 overs adding 114 runs.
Angelo Mathews, 53 not out at the start of play, could only add another 13 runs while Shaminda Eranga was the only Sri Lankan batsman to delay the hosts for any length of time.
The No 10 struck an unbeaten 45 off 62 balls as the visitors managed to reach 407.
Tim Southee took three of the final five wickets to fall and finished with figures of 4-91, to add to the two wickets he claimed in the first innings.
Set a victory target of 105, New Zealand slipped to 43-2 before Kane Williamson and Ross Taylor steered them over the finishing line without too many problems.
Taylor???s unbeaten 39 came off 63 deliveries and yielded half-a-dozen fours, with skipper Brendon McCullum saying it has been a good year for the Black Caps.
After a dismal 2013, which saw the Kiwis dismissed for 45 by South Africa during a run of nine Tests without a win, New Zealand bounced back in style during these past 12 months.
In 2014, they won five and drew two of their nine Tests and McCullum admitted: “We sat down and worked out where we were at after being bowled out for 45.
“We knew we had to make some changes, not just from an internal point of view but also externally and how we are viewed by the public.
“It was essential to all of us that we wanted the public support and the best way to go about that is to make sure you’re out there and trying to represent New Zealand with everything you’ve got.
“There is always pressure on places even when you achieve a record five Test wins in a calendar year.”
The second and final Test against Sri Lanka starts in Wellington this Saturday.