The hosts swept the two-match series in Galle 2-0 with their 100th Test win and maintained the tourists' 100 percent losing record in cricket's longest format.
Harry Tector fought a lone last-day rearguard action to try to bat out a draw, scoring 85 and mounting a 41-run stand for the ninth wicket, but Mendis took 5-64 as Ireland were bowled out for 202 in their second innings.
Asitha Fernando bowled Tector with a yorker to end the resistance and accounted for last man Ben White with the very next ball to wrap up another emphatic win in style.
Sri Lanka won the first Test by an innings and 280 runs.
Ireland achieved Test status in 2017 but have now lost all their six matches in the format.
"We are very happy to take the game to day five," skipper Andy Balbirnie told reporters in Galle.
"I would like to think that we have made improvement. We need to keep doing that."
The visitors posted 492 in their first innings - among the highest-ever scores to be followed by an innings defeat.
After Sri Lanka declared on 704-3, Ireland resumed their second innings on 54-2, hoping to hold on for a draw on a track still good for batting.
But Jayasuriya dismissed Paul Stirling in the fourth over of the morning to become the quickest-ever spinner to take 50 wickets, in just seven Tests.
Fellow left-armer Alf Valentine of West Indies had taken eight games to reach the milestone more than seven decades earlier.
Mendis removed first-innings centurion Curtis Campher when he swept one straight into the hands of Kusal Mendis at leg-slip, and took his third wicket of the innings when Balbirnie drove one straight to Angelo Mathews at mid-off on 46.
Mendis then dismissed Andy McBrine and Graham Hume in successive overs to claim his fourth five-wicket haul.
It had been a "very batting-friendly" wicket, said winning captain Dimuth Karunaratne, adding he was "really impressed with the way the bowlers went about their business".
For Ireland, their next Test will be another daunting task, against England in June.
"We have a Test match coming up at Lord's soon and these experiences of playing three Tests inside a month will come in handy," said Balbirnie, whose side were also well beaten by Bangladesh.
"This has been a great learning curve."
