Chasing 275 for victory, New Zealand were all out for 211 on Monday despite Rachin Ravindra's gallant 92, their batters unable to negotiate the sharp turn and low bounce at Galle International Stadium.
Jayasuriya (5-68) claimed his eighth five-wicket haul in Tests, with seven of them coming at his happy hunting ground in Galle.
New Zealand began the final day needing 68 runs for a memorable win with two wickets in hand, but Sri Lanka took only four overs to wrap up victory.
The tourists' slim chances evaporated in the second over of the day when Jayasuriya ended Ravindra's stellar knock, which included nine fours and a six.
The batter reviewed the lbw decision more in desperation than anything else, and replays confirmed the New Zealander was directly in front.
Jayasuriya bowled William O'Rourke in his next over to seal Sri Lanka's victory, beating the batter's defensive prod at an arm ball.
Jayasuriya's nine wickets spread over both innings earned him the player of the match award, while Ramesh Mendis claimed three second-innings wickets.
The second Test, which is also in Galle, begins on Thursday.Â
