When rain ended the third's play at 4.15pm, England were 8-224 in their second innings - a lead of 360 - with Moeen 67 not out from just 59 balls.
When Moeen Ali walked to the crease at 134-6, England's lead was 270 and the Proteas were sniffing an unlikely win and a 2-2 draw.
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The captain followed soon after tea, edging Anderson behind, and Kagiso Rabada fell to the same bowler, who was harbouring hopes of a first Test five-for on his home ground.
The corresponding South Africa record is 414 for four - the second-highest fourth-innings total to win a Test of all-time - against Australia at Perth in 2008. The second of those, a straight drive that took him to a 49-ball 50, was "caught" in the England dressing-room balcony by Jonny Bairstow, much to the delight of the crowd.
After establishing a healthy first-innings lead of 136, England lost three wickets either side of lunch.
Amla took three fours from one Roland-Jones over to signal South Africa's intent to fight back after the interval and brought up his fifty by guiding Ben Stokes down to fine leg for another boundary.
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Olivier had his tail up, causing problems with his clever right-arm medium-fast deliveries and, after troubling Stokes outside the off-stump on several occasions, got his reward when the England batsman edged to skipper Faf du Plessis at first slip.
Starting off the day on 220/9, South Africa only added six runs to their overnight score before they were bowled out when Duanne Olivier was caught behind off the bowling of Stuart Broad for four runs.
Alas, for the Proteas, Morkel's magic was outshone by the marvellous Moeen and thus the series was encapsulated as South Africa were ultimately left to contemplate a likely defeat and rue what might have been.
"We've put pressure on them and the match has turned our way", said Moeen. "I was desperately disappointed not to chalk up another hundred in Test cricket".
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