Chasing 275 for victory, New Zealand were all out for 211 despite Rachin Ravindra's gallant 92, their batters unable to negotiate the sharp turn and low bounce at the venue.
Sri Lanka spinner Jayasuriya (5-68) claimed his eighth five-wicket haul, seven of them coming at his happy hunting ground in Galle alone.
"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."
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.
New Zealand's 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 Ravindra was plumb 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," New Zealand 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."
Jayasuriya's match haul of nine wickets 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, also in Galle, begins on Thursday.
