South Africa were dismissed for 169 - the highest total so far of a low-scoring match - in their second innings on the fourth day, with no Proteas batsman making more than captain Dean Elgar's 36.
England skipper Ben Stokes led the hosts' attack with 3-39, while Stuart Broad took 3-45.
Ollie Robinson's return of 2-40 in 15 overs meant the Sussex seamer had taken 51 wickets in his 11 career Tests at a miserly average of 19.80.
Only one England bowler since the end of the Second World War has taken more wickets at a better average – express quick Frank Tyson, whose 76 Test wickets cost just 18.56 apiece.
Victory would mean England had won six of their seven Tests since a new leadership duo of Stokes and coach Brendon McCullum took charge at the start of this season.
The climax to a three-Test series, currently level at 1-1, has effectively been reduced to a three-day game.
Thursday's opening day was washed out without a ball bowled before Friday's play was abandoned following the death of Queen Elizabeth II.
