The hosts dismissed South Africa for 179 on the third day thanks to some outstanding seam bowling from captain Ben Stokes, Ollie Robinson and James Anderson.
England reduced South Africa to 54 for 3 before an obdurate stand of 87 between Keegan Petersen and Rassie van der Dussen held England up before Stokes removed both after tea.
With the second new ball, South African collapsed by losing five wickets for seven runs in 30 balls with Robinson ending with 4 four 43 and Anderson 3 for 30.
On the opening morning of the test South Africa won the toss and elected to bat first. The visitors were soon in trouble when Sarel Erwee(3) was caught behind off Anderson.
Wickets fell at regular intervals with South Africa eventually dismissed for 151 with tail ender Kagiso Rabada top scoring with 36 and Anderson and Broad picking up three wickets apiece.
England were soon in trouble at 34 for 3 but were rescued by Jonny Bairstow who made 49 and centurions Stokes 103 and Ben Foakes with an undefeated 113 as England declared their first innings on 415 for 9, an overall lead of 264.
Robinson and Anderson removed openers Erwee(25) and Elgar(11) to put the hosts in a good position to push for victory.
Despite 42 from Petersen and 41 from van der Dussen, South African were always struggling before England wrapped up victory late into Saturday afternoon on the third day to set up a series decider at the Kia Oval starting on the 8Th September.
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