Jayasuriya takes 6-42 as New Zealand collapse to 88 all out
Sri Lanka captain Dhanajaya de Silva enforced the follow on and New Zealand stuttered to lunch at 3-1 after facing two overs in their second innings, still needing 511 more to avoid an innings defeat.
Tom Latham was the man out before the interval. He departed without scoring, caught by Pathum Nissanka at short leg off the bowling of Nishan Peiris who earlier had taken three first-innings wickets.
Jayasuriya, who began the Test with 88 wickets from 15 matches, now needs six more in the second innings to equal the record for fastest bowler to 100 Test wickets.
The record has stood since 1896 when England's George Lohmann reached 100 wickets in his 16th Test.
Resuming at a precarious 22-2 in their first innings, New Zealand's batsmen barely put up a fight and their highest partnership was the 20 added by last pair Mitchell Santner, who top scored with 29, and William O'Rourke (2).
Leading batsman Kane Williamson could not tame Sri Lanka's formidable spin attack as Jayasuriya dismissed him in just the third over to trigger the collapse.
Captain de Silva took five catches at first slip to tie the record for most in a Test innings by a non-wicketkeeper, a milestone shared with 15 others.