Chasing 275 for victory, the Kiwis were all out for 211 on Monday despite Rachin Ravindra's gallant 92, their batters unable to negotiate the sharp turn and low bounce.
Jayasuriya (5-68) claimed his eighth five-wicket haul in Tests, with seven of them coming at his happy hunting ground in Galle.
"The win in England boosted our morale," home captain Dhananjaya de Silva said referring to their Test match victory at The Oval earlier this month.
"We've the ability to win Tests and the boys are delivering now."
Prabath Jayasuriya's five-wicket haul scripts a memorable Sri Lanka win over New Zealand �— ICC (@ICC) 🇱🇰 go up 1-0 in the series 🔥#WTC25 | #SLvNZ �: https://t.co/PHqmvlAFRP pic.twitter.com/xGbPuc7B7lSeptember 23, 2024
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 caught 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.
"Tough place to come but a good Test match throughout," Black Caps captain Tim Southee said.
"There's plenty of positives for us to take away from it but obviously disappointing to come out on the wrong side."
William O'Rourke is clean bowled, bringining an end to the first Test in Galle. (AP PHOTO)
Jayasuriya's nine wickets spread over both innings earned him the player-of-the-match award, while Ramesh Mendis claimed three second-innings wickets.
"I like this Galle pitch," Jayasuriya said.
"I just concentrate on bowling a consistent line and length, focus on that and get the results."
The second Test, which is also in Galle, begins on Thursday.
