Sri Lanka spinner Lasith Embuldeniya claimed seven wickets to help bowl out England for 344 as the tourists conceded a first-innings lead of 37 runs on day four of the second Test on Monday.
Dilruwan Perera trapped overnight batsman Jack Leach lbw for one on just the 11th ball of the day after England resumed on 339 for nine in Galle.
Leach took a single off Perera and No 11 James Anderson hit a boundary off Embuldeniya, who returned career-best figures of 7-137 in just his ninth Test, to add five runs to the overnight score.
Skipper Joe Root scored 186 - after his double century in the first Test - to steer England's response to Sri Lanka's first innings total of 381.
He was run out in the final over on Sunday off a direct throw from short-leg fielder Oshada Fernando.
Riding on Root's 228, England won the opening Test to lead the resumed series, which started last March but was postponed due to the coronavirus pandemic.