Jayawardene, who gave up the captaincy earlier this year to concentrate on his batting, compiled his 26th Test century on the easy-paced wicket at the Galle International Stadium.
Dilshan hit the fastest Test half-century by a Sri Lankan off 30 balls, upstaging former captain Arjuna Ranatunga's 31-ball effort against India at Kanpur in 1986.
Jayawardene put on 118 for the third wicket with Dilshan and 159 for the undefeated fourth with Thilan Samaraweera (82 not out) after New Zealand seamer Chris Martin struck twice in his first two overs.
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Martin gave the Test a dramatic start when he had Tharanga Paranavitana, Dilshan's fellow opener, edging the third ball of the match to wicket-keeper Brendon McCullum.
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Martin then removed Kumar Sangakkara for eight, the S