In Sri Lanka's biggest win as well as Ireland's worst defeat to date, the visitors were bundled out for 168 on day three in a humid seaside Galle stadium after being made to follow on.
"We fought in stages but not enough. We are learning what it means to play Test cricket," Ireland captain Andrew Balbirnie said.
"Hopefully we'll learn quickly and put up a better fight in the next game."
"I know this is our biggest Test win but what matters to me is winning Test matches and not the margins," Sri Lanka skipper Dimuth Karunaratne said.
"(Ireland) are new to this format and they are still learning about this format. But I think they were really good and in future, they will be a good team as well."
It was the 12th-largest innings victory in Test history, behind England's win over the West Indies in 2007 by an innings and 283 runs.
Sri Lanka's previous biggest win was when they hammered Zimbabwe by an innings and 254 runs in 2004 in Bulawayo.
Ireland have now lost all five of their matches since securing Test status in 2017. Their previous biggest defeat was to England by 143 runs in 2019.
In the first Test between the two sides, Sri Lanka made Ireland's fielders toil from the start, racking up 591 for six in their first innings before declaring on day two.
Karunaratne top scored with a masterful 179 while Kusal Mendis posted 140. Dinesh Chandimal and Sadeera Samarawickrama then hit unbeaten hundreds.
Sri Lanka then had Ireland on the ropes on 4-2 before bowling them out for 143 with Jayasuriya taking a career-best 7-52 and Lorcan Tucker top-scoring with 45.
With a first-innings lead of 448, Sri Lanka enforced the follow-on, reducing them to 40-5 before lunch but Harry Tector and Curtis Campher resisted with a 60-run stand.
The partnership was broken when Nishan Madushka pulled off a good reflex catch at short leg to dismiss Campher for 30 and give Ramesh Mendis his second wicket in the innings.
Mendis finished with four wickets, becoming after his 11th appearance the joint-fastest Sri Lankan to get 50 Test wickets alongside off-spinner Dilruwan Perera.
Tector, a tall right-handed batter, looked impressive with solid defence despite copping a nasty blow on the ribs from an Asitha Fernando delivery.
But his promising innings was cut short when he pushed one to mid-off and hurried for a single but there was no response from the non-striker.
Midway down the pitch, he turned back, slipped and despite a dive couldn't save himself, departing for 42.
Jayasuriya bagged another three scalps, including the final one to fall, trapping Ben White leg-before for one.
"All I try to do is to keep things simple. Bowl to good lines and lengths and don't offer too many loose balls and create pressure by bowling in the right areas," Jayasuriya said.
"When you do that you get wickets. Galle has been a great venue for me."
