Sri Lanka took just four overs to bowl the Black Caps for just 211 in their second innings to score a 63-run win in Galle.
Sri Lanka now lead the two-match series 1-0, with the second test to come starting on Thursday, also in Galle.
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Prabath Jayasuriya took the two wickets required to end with 9-204 overall.
In the hope of chasing down a further 68 runs, Rachin Ravindra moved from 91 to 92 but was trapped around the wicket as Jayasuriya tossed one up on leg and he played down the wrong line.
He reviewed, but once it was confirmed to be pitching in line, there would be no saving him.
And any lingering hopes of a famous rearguard were then swiftly quashed, as Will O'Rourke kept out just five deliveries before the sixth slipped past his forward defence and on to off stump to hand Jayasuriya his fifth wicket of the innings.
O'Rourke would not quite be satisfied with how the match ended after an impressive showing, as he finished with eight wickets - the second-most across the test, and the same as Ajaz Patel, who was left two not out.
Sri Lanka have moved to third place on the World Test Championship table, just ahead of New Zealand.
The Black Caps have made their task considerably tougher, with series against India and England coming up in the next few months.
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