Sri Lanka spinner Rangana Herath earned the praise of both captains after his man-of-the-match performance in Galle.
The left-arm spinner took 6-43 off 18 overs as New Zealand, trailing by 26 runs, crashed to 118 in their second innings after losing their last nine wickets for 83.
Sri Lanka easily chased down a 93-run victory target, with debutant Dimuth Karunaratne unbeaten on 60 and Tharanga Paranavitana 31 not out.
"Rangana knows what he is doing with the ball and he knows what the batsman is doing as well," said his skipper Mahela Jayawardene.
"After Murali (Muttiah Muralitharan), from what I've seen in Sri Lankan cricket, he is the next best thing. Others have chipped in but Rangana has become the leading bowler for us.
"Because of the pressure he is creating on the opposition and the way he's performed, it has enabled us to be consistent in Test cricket in the last two years or so."
The Test was evenly poised at stumps on Sunday when New Zealand were nine runs ahead with as many wickets in hand, but Herath turned the match in Sri Lanka's favour with four quick wickets in the morning.
New Zealand captain Ross Taylor admitted Herath was the difference: "He dominated our batting line-up in both innings. He is a very good bowler at the moment and the leading wicket-taker in 2012.
"He is very consistent in this part of the world and we have got to come up with a way of negating his wicket-taking ability. We have to pick ourselves up and give ourselves a chance in the next match."