The tourists collapsed to 93-5 before tea, but had recovered to 211-5 when bad light forced an early close.
That left Sri Lanka 114 runs adrift of an England first-innings total of 325 built on Ollie Pope's 154 -- his first century as England captain.
De Silva was 64 not out and Kamindu Mendis 54 not out in an undefeated stand of 118 as Sri Lanka looked to salvage some pride at 2-0 down in a three-match series.
For left-handed batsman Mendis, currently boasting a staggeringly high Test batting average of over 85, this was the seventh time in six matches at this level the 25-year-old had passed 50 -- a run that also includes three centuries.
De Silva, however, should have been out for 23 but Test debutant Josh Hull dropped a simple catch at mid-on, the ball bursting through the towering fast bowler's giant hands following a miscued drive off spinner Shoaib Bashir.
Sri Lanka had gifted England an early breakthrough when Dimuth Karunaratne (nine) was run out by Olly Stone's direct hit from short cover after Pathum Nissanka set off for a non-existent single.
Nissanka completed a fine fifty when he punched the 20-year-old Hull for four through mid-off, his seventh boundary in 40 balls.
But it was not long before the collapse continued.
Stone, in just his second Test in three years after the injury-plagued paceman returned in England's 190-run win at Lord's last week, had veteran all-rounder Angelo Mathews (three) edging to Pope in the gully.
Hull success
The 6ft 7 inch (2 metres) 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.
