Pathum Nissanka led the way with a sparkling 53 not out, making hay in the late evening sunshine after England had earlier been put to the sword under floodlights during a chaotic third day at the Oval.
The visitors seem set to claim victory on Monday, barring yet another twist in a test match full of them.
England swiftly squandered a first-inning lead of 62 with a tired display of top-order slogging, and then almost redeemed themselves thanks to one of their new stars.
That they even had 218 to defend was down almost solely to Jamie Smith, one of England's discoveries of the summer, as he blasted an astonishing 67 to paper over the earlier capitulation.
Coming in at a perilous 69-5, after some fine Sri Lankan swing had wrecked England's innings, Surrey's Smith took one Milan Rathnayake over for 20 on his way to a superb half-century in front of his home crowd.
Ben Duckett (7), Ollie Pope (7), Dan Lawrence (35) and Joe Root (12) earlier all fell swinging, sacrificed at the altar of the relentlessly aggressive philosophy that has brought England so much success in recent years while antagonising their more traditionally-minded fans.
As much as England played some loose shots, Sri Lanka bowled well, finding consistent movement through the air after lunch.
Vishwa Fernando took the prize wicket of Root, rapping him on the boot with an in-swinger, and then doing the same to Harry Brook.
Once Smith holed out to midwicket, the tail could add only a few boundaries to leave the hosts well short of a commanding total as the sun came out and batting conditions improved at just the wrong time for England.
England had begun the day in the ascendancy, briskly removing the overnight pair of Dhananjaya de Silva and Kamindu Mendis.
Their latest surprise test selection, the 6'7" (2.01m) left-arm seamer Josh Hull, had Sri Lanka's captain de Silva caught hooking in the deep and later took the third wicket of his fledgling test career, trapping Vishwa Fernando lbw.
Chris Woakes between those two wickets had seized the prize scalp of Mendis, Sri Lanka's best batsman of the series, with a fine ball that seamed away from him and found the edge.
Olly Stone removed the dangerous tail-end hitter Rathnayake before Shoaib Bashir mopped up Asitha Fernando, to give the hosts what seemed at the time a useful first innings lead.
