Having declared their first innings on a mammoth 5-602 on Friday, Sri Lanka bundled out New Zealand for 88 in Galle on Saturday, enforced the follow-on and reduced the tourists to 5-199 in their second innings on a frenetic 13-wicket day.
Tom Blundell (47no) and Glenn Phillips (32no) had added 78 runs for the sixth wicket when bad light stopped the third day's play, leaving New Zealand 315 behind and facing an innings defeat.
After the visitors resumed at 2-22 on Saturday, Jayasuriya (6-42) and Peiris (3-33) wreaked havoc and New Zealand were bundled out inside 40 overs.
Of the three touring batters who crept into double digits, Mitchell Santner's 29 was the highest score by a New Zealander in their first-innings capitulation.
Left-arm spinner Jayasuriya registered his ninth five-wicket haul in Test cricket, eight of which have come in Galle.
Home captain Dhananjaya de Silva spent a busy day at slip taking five catches, all off the bowling of Jayasuriya.
Kane Williamson (7) added one run to his overnight score before perishing in the slips, and soon the wheels came off for the tourists.
With a first-innings lead of 514, Sri Lanka saw no need to bat again and enforced the follow-on, offering the Kiwi batters no respite from their trial by spin.
Tom Latham fell for a duck in the first over of their second innings from Jayasuriya, but Devon Conway (61) combined with Williamson (46) in a 98-run stand before the home spinners struck again.
De Silva then removed Conway and Peiris dismissed Williamson, and New Zealand slumped to 5-121 when Peiris bowled Rachin Ravindra for his sixth wicket of his debut Test.
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