England extended their lead to 145 after bowling South Africa out for 361 despite contrasting half-centuries from Quinton de Kock and Vernon Philander on day three of the first Investec Test at Lord's.
Joe Root told Sky Sports that he was in "a dream world" after he scored an unbeaten 184 in his first innings as England Test captain.
Off-spinner Ali, who on Thursday completed both halves of the Test all-rounder's double of 2,000 runs and 100 wickets, had Bavuma (59) taken at slip by Stokes via Bairstow's thigh. Yet the inexperienced pair of Temba Bavuma and Theunis De Bruyn stopped the dam bursting by adding 99 for the fifth wicket, before Jimmy Anderson returned from the Nursery End and had De Bruyn caught behind for 48.
The South Africans did succeed in taking some time out of the game, batting until nearly the halfway mark of the day's play.
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However he did not take the field before the tea interval, leaving South Africa with just two frontline fast bowlers as England extended their lead in the match.
After de Kick delivered the blitz, Philander came up with a calculated innings, realising 52 from 86 balls with seven fours before becoming Moeen Ali's fourth scalp.
After a frantic morning session in which England lost their last five wickets while adding 101 to their overnight total - Broad and Moeen scoring 83 of them - the pace slackened in the heat of the afternoon.
Just as England were gathering momentum, Kagiso Rabada (3-123) castled Ali in the 101st over and two deliveries later, trapped Mark Wood (0) LBW as the hosts were reduced to 413 for nine.
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They will at least welcome back Test captain Faf du Plessis for the Trent Bridge match after he missed the opener following the birth of his first child.
The 90th over also saw him hit three fours off Broad - the best a whipped drive through mid-on from a ball pitching outside off stump - before his exit ended a stand of 66 in 75 balls with Philander.
"I rode my luck especially early on - I didn't get my feet going, they felt stuck at times, and South Africa bowled very well and made it hard".
Moeen was responsible for the important scalp of captain Elgar as he posted figures of 2-35, but Bavuma and De Bruyn steadied the ship after a trio of less than convincing batting displays from Kuhn, Hashim Amla (29) and Duminy.
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Broad struck eight fours as well as his two sixes.
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