The pace bowler claimed five more wickets in Saturday's session to send the tourists packing for 119, finishing with figures of 10-45 in the first Test of the summer.
Rain delayed play for three hours, but England made no mistake when play resumed and wrapped things up soon after tea in a battle against the weather.
Kusal Mendis top-scored for an outclassed Sri Lanka side with 53, meaning no repeat victory for the Lions two years on from their last visit in 2014.
There was to be no reprieve for Sri Lanka, who ended their second innings on 119-10 to pave the way for an easy win for England.
The second Test of the three-match series gets underway in Durham next Friday.