England, 1-0 up in the four-test series, had the hosts in trouble before being held up by a valuable unbeaten eighth-wicket stand of 42 between Chris Morris (26) and Kagiso Rabada (20).
New South Africa captain AB de Villiers won the toss and chose to bat despite overcast conditions that offered hope for England’s seam attack.
His decision appeared to be vindicated as the hosts saw off the early threat and reached 73 for one at lunch, losing opener Stiaan van Zyl (21) when he top edged Stokes to give wicketkeeper Jonny Bairstow one of four catches.
But England struck twice in quick succession in the second session to dismiss Hashim Amla and Dean Elgar as they approached half centuries to put a brake on the home side.
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Amla fell to a vicious in-swinging delivery from Finn, caught behind for 40 not long after Elgar (46) got a thin edge to Bairstow off the spin of Moeen Ali.
An attacking innings from De Villiers ended shortly after tea as he went for 36 off 40 balls, followed by Faf du Plessis who holed out to deep square leg off Finn with a rash shot.
De Villiers, replacing Amla as captain and promising a more aggressive approach, gloved a ball from Stokes down the leg side to be caught behind just when he looked capable of producing another big score on a ground where he has had past success.
Temba Bavuma, who scored a century in the drawn second test, was then run out for 23.
Wicketkeeper Dane Vilas, who made a last-minute dash from the coastal city of Port Elizabeth to play after being called up as a replacement for the injured Quinton de Kock, struck some lusty boundaries before pulling the ball to deep backward square.
Moeen caught the high ball to give Stuart Broad his first wicket as Vilas perished for 26.
“We had a harsh first two sessions with the bat but 300 in the first innings will put us in a good position,” Elgar told reporters.
“England’s bowlers were spot on and didn’t give us a lot of scoring opportunity. But when the tail wags it’s frustrating and we hope we can capitalise on that.”
England bowling coach Otis Gibson said he expected the first half-hour of Friday’s play to be “full of energy”.
“We’ve got to come out tomorrow and be very aggressive," he said.
(Editing by Justin Palmer; mark.gleeson@thomsonreuters.com; +27828257807; Reuters Messaging: Reuters Messaging: mark.gleeson.thomsonreuters.com@reuters.net)
