Sri Lanka, already trailing 2-0 in the three-match series, found themselves 183 runs behind as they left the field. England had posted a total of 325 in their first innings, with Ollie Pope scoring an impressive 154 runs to secure his first century as captain of the England team.
Fast bowler Stone took 2-28 in five overs, with Sri Lanka indebted to Pathum Nissanka for the opener's 64.
One consolation for Sri Lanka was that the in-form Kamindu Mendis, averaging over 50 in the series before this match, was 34 not out, with captain Dhananjaya de Silva unbeaten on 16.
Sri Lanka gifted England a breakthrough when Dimuth Karunaratne was run out by Stone's direct hit from short cover after Nissanka set off for a non-existent single.
A brief moment of farce followed as the umpires, concerned about the light, even with the floodlights on, insisted paceman Chris Woakes complete the over bowling spin.
At the end of the over, the officials ruled conditions had brightened sufficiently for England's fast bowlers to bowl at full pace.
The recalled Kusal Mendis was left to rue that decision as he edged Woakes to second slip on 14, with Sri Lanka now 70-2.
Nissanka completed a fine fifty when he punched 20-year-old England debutant fast bowler Josh Hull for four through mid-off, his seventh boundary in 40 balls faced.
It was not long before the collapse continued.
The injury-plagued Stone, in just his second Test in three years after he returned in England's 190-run win at Lord's last week, had Angelo Mathews out for three when the veteran all-rounder edged an 85 mph delivery to Pope in the gully. - Hull success -
The 6ft 7 inch tall Hull then took his first wicket in Test cricket, the towering left-hander removing Nissanka when the opener's drive was well caught by a diving Woakes at cover.
And 91-4 became 93-5 when Stone had Dinesh Chandimal lbw for a duck.
But left-handed batsman Kamindu Mendis struck five fours, including an elegant square-drive off Gus Atkinson from the last ball before tea.
Earlier, England lost their last six wickets for 35 runs after resuming on 221-3.
Pope was 103 not out, after managing a mere 30 runs in four previous innings since succeeding the injured Ben Stokes as skipper at the start of this series.
Sri Lanka's wayward four-man pace attack had failed to make the most of overcast conditions and a green-tinged pitch after De Silva won the toss.
Their performance improved on Saturday, fast bowler Milan Rathnayake finishing with innings figures of 3-56 in 13.1 overs.
Left-arm quick Vishwa Fernando removed Pope, when the 26-year-old right-hander, on his Surrey home ground, hooked to deep square leg to end a 156-ball innings including 19 fours and two sixes.
England, who swept the West Indies 3-0 earlier this season, are chasing their first home Test clean sweep since 2004, when Michael Vaughan oversaw seven successive wins.
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