Captain Alastair Cook was 67 not out for his first half-century of the series and led England to stumps on the second day at SuperSport Park.
England has already won the series with a 2-0 lead but has the chance to become the first touring team in 10 years — and only the second since South Africa's return to test cricket in 1992 — to win three tests in a series in the country.
South Africa's first-innings total at Centurion was built on three centuries, by opener Stephen Cook on his debut, by recalled wicketkeeper Quinton de Kock, and a 25th career ton by Hashim Amla.
England contributed to its big deficit as well, dropping all three centurions: Cook twice, Amla when he was only on 5, and De Kock three times on his way to a rollicking 129 not out from 128 balls.
"We'll look back on this game and realize we let them off the hook massively," said England's Ben Stokes, who missed the first chance off De Kock early in the day when he was on 28.
But England was also still fighting at the end of the series, with Alastair Cook finding his timing for the first time in the series to caress 10 fours to the boundary, and Joe Root 31 not out alongside his skipper at stumps on the second day.
England trailed by 337 but, on a true pitch that favors the batsmen, the tourists have a good chance to gain parity and maybe even push ahead with their deep batting lineup.
"Cookie's obviously struggled this tour. So when he did get that 50, it was almost like a hundred. We were so pleased for him," Stokes said.
Cook needs another 50 runs to be the first Englishman to the significant milestone of 10,000 test runs. He would be only the 12th player to achieve that feat. But his first mission is to help England get level at Centurion.
Even if another win isn't possible for the tourists, England can become the first team since Ricky Ponting's Australia in 2005-06 to prevent South Africa winning a test in a home series.
Cook settled England after opening partner Alex Hales went for a breezy 15 to a square drive he sliced straight to Dane Piedt at point. Nick Compton's cautious 19 from 52 balls was ended by a ball that misbehaved by keeping surprisingly low to trap him lbw. Young fast bowler Kagiso Rabada claimed both those wickets.
England was 22-1 and 78-2.
"We got through a tough patch there as a batting unit," Stokes said. "Spending a lot of time in the field there can be some tired bodies and some tired minds. We've done well."
Stephen Cook's 115 and Amla's 109 had set up South Africa's innings on the first day, but De Kock's bullish hundred from No. 7 on the second day really gave the home team some big momentum.
Recalled for the second test, only to then get injured for the third game, the young left-hander showed his potential with a flourish of boundaries. He crashed 17 fours and two sixes, with the first of those two maximums nearly ending up in the swimming pool in a fan enclosure down by the mid-wicket boundary.
His maiden test century, the second maiden ton of the South African innings after Cook's, came when he squeezed a ball out to cover and took advantage of a misfield to scramble through for the single.
"It's quite relieving," De Kock said. "I'm sure you can ask any person ... they're always going to tell you the same thing. But it's just the start."
That run to go to 100 was one of few singles for the free-hitting De Kock.
On 25 overnight, he scored 60 off 63 balls in the first session and then clattered another 44 off 34 deliveries after lunch.
Stokes, who finished with 4-86 to lead England's bowling attack, stopped the damage when he had last man Morne Morkel out lbw for a duck, leaving De Kock stranded at the other end.
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