Jayasuriya grabs five wickets as Sri Lanka win first New Zealand Test
Set a target of 275 to win, the Black Caps were bowled out for 211 less than 15 minutes after play resumed, with Rachin Ravindra top-scoring on 92.
"When you come to Galle, all that you need to do is to stick to the basics," said man-of-the-match Jayasuriya.
"The wicket provides plenty of assistance and you just need to bowl with discipline. Your consistency is the key and there was lot of help from the rest of the spinners as well."
Ravindra had finished on Sunday waging a lone battle on a sharply turning pitch in Galle, where the tourists had lost all four of their previous Test encounters.
His knock was the highest score by a New Zealander at the ground, eclipsing Ross Taylor's 89 in 2019.
But the slender path to victory he had opened was snapped shut when Jayasuriya trapped him lbw while trying to defend on the back foot.
The decision was upheld despite Ravindra's review, and he departed having added just one run to his overnight total.
Jayasuriya, who took nine wickets across the match and finished the second innings five for 68, brought the match to an end when he bowled rookie pacer William O'Rourke for a duck.
"Disappointed we didn't get enough lead in the first innings," New Zealand skipper Tim Southee said.
"We had the game in the hand, but we threw it away."
'Sensational' O'Rourke lauded
O'Rourke, 23, had impressed for New Zealand with eight wickets, including his second five-wicket haul from just three Test appearances in the first innings.
"William O'Rourke has been brilliant for us," Southee said.
"For him to come here and do what he did was absolutely amazing. He's never played in conditions like these... when there is not much for pace bowling. His effort was sensational."
New Zealand had cause for cheer after fighting on for a fifth day during their first Test in six months on a pitch where they have always struggled.
Their scheduled warm-up against Afghanistan earlier in September was washed out without a ball being bowled.
Left-arm spinner Ajaz Patel claimed 6-90 in Sri Lanka's second innings, the best figures by a New Zealand bowler in Sri Lanka since Daniel Vettori's 6-64 in Colombo in 1998.
Tom Latham, Kane Williamson, and Daryl Mitchell all contributed with half-centuries in the first innings.
The second and final match of the series will be held at the same venue from Thursday.