Left-armer Thushara finished with 3-29 and Kulasekera took 2-30 as Pakistan struggled against Sri Lanka's 232-9 and were bowled out for 196 in 44.
4 overs at the Rangiri International Stadium.
The tourists, beaten 2-0 in the preceding Test series, slipped to 134-8 before tailenders Umar Gul and Mohammad Aamer boosted the total with a rollicking ninth-wicket stand of 62 runs.
Gul top-scored for Pakistan with 33 off 21 balls, including four boundaries in one over off fast bowler Lasith Malinga.
Teenager Aamer made 23.
The pair had taken the total to 196-8 when both batsmen fell in the space of two deliveries to hand Sri Lanka the lead in the five-match series.
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Aamer was run out by a direct throw from Mahela Jayawardene, before Gul was bowled by Malinga next ball.
Sri Lankan captain Kumar Sangakkara praised the depth in his team, but conceded they had almost given it away during the stand between Gul and Aamer.
"We let it get away a bit at the end, perhaps we got a bi