Pakistan, set a tough victory target of 510, were bowled out for 300 in their second innings just before stumps on the fourth day at the Galle International Stadium.
Off-spinner Suraj Randiv, who claimed four wickets in the first innings, took three more in the second to fashion Sri Lanka's emphatic win.
Pakistan's innings revolved around a 151-run stand for the fifth wicket between Younis Khan (87) and Asad Shafiq (80), before both fell on either side of the tea interval.
The second Test starts at the Sinhalese sports club in Colombo on June 30.
Brief scores:
Sri Lanka 472 and 137-5 declared; Pakistan 100 and 300.
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