The off-spinner, who became Test cricket's most successful bowler in the first innings, got out Matt Prior and Ian Bell in the space of three deliveries with the second new ball to end England's spirited challenge.
The tourists, needing to bat out the final day for a draw after being set an improbable victory target of 350, were bowled out for 261 in the last hour of the match at the Asgiriya stadium.
Sri Lanka took a 1-0 lead in the three-match series going into the second Test, which opens at the Sinhalese Sports Club in Colombo on Sunday.
England, who resumed the final day at 9-1 in their second knock, slumped to 90-5 in the morning session before Bell (74) launched a heroic rescue attempt in the company of the lower order batsmen.
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He put on 49 for the sixth wicket with debutant Ravi Bopara (34) and a defiant 109 for the seventh with wicket-keeper Prior (63) that threatened to frustrate Sri Lanka's victory bid.
Muralitharan, kept wicketless for 31 overs, struck in his second o