Left-arm spinner Rangana Herath claimed five wickets as the Kiwis, set an improbable victory target of 494 runs, were bowled out for 397 in their second innings just before tea on the fifth day.
Sri Lanka had to overcome spirited batting by Kiwi captain Daniel Vettori and heavy rain during the lunch break that delayed resumption by 35 minutes to build on their 202-run win in the first Test at Galle.
Vettori hit 140, his fourth Test century, to lead a stirring fightback by the lower order after the top batsmen had failed to put up any resistance.
The left-hander frustrated the Sri Lankans for more than four hours in which he faced 189 deliveries and hit 16 boundaries, before he was last man to be dismissed.
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Vettori scored 272 runs and claimed 10 wickets in the series, and in the process became only the eighth batsman in the world to achieve the double of 3,000 runs and 300 wickets.
Vettori put on 124 for the seventh wicket with Jacob Oram (56) and 69 for the ninth with Iain O'Brie