Broad had a spell of five wickets for one run to send South Africa crashing to 71 for eight at tea, a paltry lead of 61.
It was a performance that put England within sight of a series victory with one match to play.
England led on the first innings by 10 runs after being bowled out for 323 and the match was evenly poised as South African opening batsmen Dean Elgar and Stiaan van Zyl put on 23 runs at the start of the second innings, with Broad having conceded 13 of the runs.
Van Zyl was dropped, a difficult chance to James Anderson at third slip, and the single that was taken as the ball looped away was to prove the only run that Broad conceded while tearing the heart out of the batting.
Van Zyl was caught at gully. South African captain AB de Villiers came out to a roar of appreciation from a near-capacity crowd but was caught behind off a superb delivery which cut back sharply to take an inside edge.
Hashim Amla was superbly caught at short leg by James Taylor off a full-blooded leg glance and Temba Bavuma was bowled when he ducked to avoid a short ball which hit his glove and went on to the stumps.
But the carnage continued. Steven Finn had Dane Vilas caught by a diving Taylor at short leg and Chris Morris was bowled by a full ball from Ben Stokes.
Kagiso Rabada made a quick 16 before he was caught behind off Stokes, leaving Faf du Plessis and Hardus Viljoen to try to take South Africa to respectability.
Earlier Rabada claimed a maiden five-wicket haul to restrict England's first innings lead to 10 runs.
Rabada, 20, playing in his fifth Test match, took five for 78 and led the Souh African attack, bowling at pace with consistent control.
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