The stars of the show were the Kiwi fast bowlers, Trent Boult and Tim Southee took four wickets, while Neil Wagner and Jimmy Neesham had done the same in the first innings.
The Kiwis made 441 before bowling Sri Lanka out for 138, forced them to follow on, and bowled them out for 407. As such, they needed just 105 runs to win, which they got on day four.
Day four started with Sri Lanka 10 runs behind and with five wickets in hand. They were on 293 for five, with Angelo Mathews well set on 53 overnight. He advanced to 66 before being removed by Southee.
Shaminda Eranga's 45 took the score past 400 and brought up a tree-figure lead, but it was never going to be enough.
The Black Caps needed 30.4 overs to reach the target, losing both openers cheaply. But Ross Taylor and Kane Williamson, scoring 39 and 31 respectively, knocked off the runs to give their side an extra day's rest before the new year.
The second and final Test starts on 3 January at the Basin Reserve in Wellington.