The home team reached 264 for two at the close after Jayasuriya clubbed 12 fours and a six in his 85 from 104 balls.
Trailing England's total by just one run, Sri Lanka are comfortably placed to build a huge first innings lead when the Test resumes on Saturday.
Thilan Samaraweera and Mahela Jayawardene also toiled with the English attack to score half centuries. Samaraweera was unbeaten on 68 with Jayawardene 60 not out.
Jayasuriya put on 71 for the first wicket with Kumar Sangakkara (31) and 67 for the second with Samaraweera.
Samaraweera, dropped twice at slip by Marcus Trescothick on 12 and 46, added a further 126 runs with Jayawardene in an unbroken third-wicket partnership.
STYLISH CAMEO
Jayasuriya and Sangakkara, who opened in place of the injured Marvan Atapattu, had taken the attack to England's bowlers in the morning in a run-a-ball stand before James Kirtley broke through just before lunch.
Having stroked six boundaries in a stylish cameo, Sangakkara was caught at slip by Trescothick after edging a well-disguised slower ball.
Jayasuriya was given a life on 38 when Graham Thorpe failed to hold on to a diving one-handed catch in the gully off Andrew Flintoff, who bowled a tight spell.
Jayasuriya, who had reached his 50 just before the lunch interval, settled down after the break and picked up the majority of his runs in sweep strokes.
But within sight of his first century since the 145 he scored against Bangladesh in July 2002, the 34-year-old left-hander edged a Flintoff delivery to Trescothick at slip.
After Jayasuriya's departure Samaraweera and Jayawardene consolidated, slowly accumulating runs as England became increasingly defensive.
England added just six runs in the morning before Chaminda Vaas trapped Kirtley (one) and James Anderson (one) lbw to wrap up their first innings.
Vaas finished with three for 64 and Muttiah Muralitharan three for 40 from 40 overs, which took his series tally to 23 wickets.
Atapattu was unable to open the Sri Lankan innings after splitting the webbing between the middle and index fingers of his left hand while catching Gareth Batty on Thursday.
The first two tests were drawn.
Scorecard:
England 1st Innings (overnight 259 for 8)
M Trescothick c Jayawardene b Muralitharan 70
M Vaughan c Jayawardene b Chandana 18
M Butcher c Sangakkara b Fernando 23
N Hussain lbw b Vaas 8
G Thorpe lbw b Muralitharan 13
A Flintoff c and b Muralitharan 77
G Batty c Atapattu b Chandana 14
C Read not out 17
A Giles run out 10
J Kirtley lbw b Vaas 1
J Anderson lbw b Vaas 1
Extras (b-4, lb-8, nb-1) 13
Total (all out, 101 overs) 265
Fall of wickets: 1-78, 2-108, 3-114, 4-135, 5-139, 6-226, 7-236, 8-258, 9-259.
Bowling: Vaas 17-5-64-3, Fernando 12-3-55-1, Samaraweera 4-1-11-0, Chandana 26-7-82-2, Muralitharan 40-21-40-3, Jayasuriya 2-1-1-0.
Sri Lanka 1st innings:
K Sangakkara c Trescothick b Kirtley 31
S Jayasuriya c Trescothick b Flintoff 85
T Samaraweera batting 68
M Jayawardene batting 60
Extras (lb-10, nb-5, w-5) 20
Total (for 2 wkts in 82 overs) 264
Fall of wickets: 1-71, 2-138.
Bowling: Kirtley 14-3-55-1, Anderson 11-3-40-0, Flintoff 2-0-33-1, Giles 23-6-63-0, Batty 19-3-48-0, Vaughan 1-0-5-0, Trescothick 2-0-10-0.
