Pope had managed a mere 30 runs in four previous innings since succeeding the injured Ben Stokes as skipper at the start of this series.
But when bad light ended play for the day, Pope was a run-a-ball 103 not out, with England well-placed on 221-3.
Sri Lanka, already 2-0 down in this three-match series, failed to make the most of overcast conditions and a green-tinged pitch after captain Dhananjaya de Silva won the toss.
And England should arguably have had two century-makers on Friday with opener Ben Duckett, not for the first time, giving his wicket away after making 86.
A rare bright spot for struggling Sri Lanka was the dismissal of Joe Root for 13 shortly before tea.
Root was fresh from two hundreds in a 190-run win at Lord's where he set a new England record of 34 Test centuries.
England, who swept the West Indies 3-0 earlier this season are chasing their first home Test campaign clean sweep since 2004, when Michael Vaughan oversaw seven successive wins.
Duckett was soon into his stride, with the left-hander cover-driving fours off successive Milan Rathnayake deliveries.
Lawrence out cheaply
It was a different story for makeshift opener Dan Lawrence, a middle-order batsman by trade.
Lawrence was forced to face the new ball this series in the absence of the injured Zak Crawley.
He had made just five when he got in a tangle against a short-pitched Lahiru Kumara delivery and top-edged a simple catch to gully.
Number three Pope got off the mark in style by cutting a loose ball from fast bowler Rathnayake for four and he also hooked a six off Lahiru Kumara.
The 29-year-old Duckett completed a speedy fifty off just 48 balls, including seven fours.
Even with the floodlights on, the umpires decided it was too dangerous to continue and stopped play for bad light, with England 76-1 off 15 overs. Rain then fell as well and the match did not resume until 15:10 BST.
It was not long before Duckett ramped Kumara for six over fine leg and uppercut him high over third ma, with all of Sri Lanka's four-man pace attack struggling to maintain a challenging line and length.
But Duckett, in sight of just his fourth century in 26 Tests, was out when he miscued an extravagant scoop off Rathnayake to wicketkeeper Dinesh Chandimal.
It was a careless end to a stand of 95 in 16 overs with Pope.
The 26-year-old Pope was fortunate with a top-edged six off Kumara but the persevering paceman did have Root hooking to fine leg, where Vishwa Fernando held onto the catch despite slipping on the wet turf.
Surrey favourite Pope, 84 not out at tea, went to his seventh hundred in 49 Tests, but first at the Oval, when he stylishly square-drove Asitha Fernando for his 13th four in 102 balls, also including two sixes.
Soon afterward bad light at 17:53 BST again saw the players leave the field, with no more play possible Friday.
