Swann and Broad took three wickets each to leave the hosts reeling at 76 for six in their second innings, trailing England by 156 runs with one day left.
Ian Bell made a fine 141 to lead the touring side to a huge total of 575 for nine declared and off-spinner Swann dismissed Ashwell Prince and Hashim Amla before tea.
Jacques Kallis, AB de Villiers, JP Duminy and Graeme Smith then fell in rapid succession after the interval and only a gritty stand between Mark Boucher and Morne Morkel enabled South Africa to reach the early close without further loss.
Smith and Prince survived the opening burst in relative comfort but Swann’s second delivery deceived Prince (16) by going straight on, catching the inside edge of the bat and rebounding off the pad to short-leg where Bell took a low reflex catch.
The off-spinner then bowled Amla for six as the batsman drove loosely and the ball turned sharply through the gate.
Broad struck three times after tea when Kallis (three), De Villiers (two) and JP Duminy (nought) all made misjudgments trying to leave the ball.
Kallis shouldered arms and had his off stump knocked over by an in-swinger which also seamed off the pitch.
De Villiers left an in-swinger and was given out lbw after asking for an umpire decision review (UDR) and Duminy lasted one delivery, withdrawing his bat too late and deflecting the ball on to his stumps.
STRAIGHT DELIVERY
Smith fell three overs later when he played across a straight delivery from Swann and was adjudged lbw, the video review showing the ball would have clipped the leg stump.
Swann had claimed three for 22 in 12 impressive overs when bad light stopped play with 17 overs remaining and Broad gave a wonderful display of controlled swing bowling in taking three for 18 from nine overs.
England had resumed in the morning on 386 for five and South Africa claimed just one wicket in the session, Matt Prior falling for 60.
Bell, whose place in the team has been under great scrutiny, continued to bat with immense focus and energy, stroking 10 fours and a six in his ninth test century.
Bell and Prior extended their sixth-wicket partnership to 112 before Prior was bowled trying an ambitious cut at off-spinner Duminy.
England declared nine overs after lunch, Bell finishing as top-scorer with a pugnacious 141 before falling to Dale Steyn with the third new ball.
Morkel, with three for 78, and Duminy, three for 89, were the pick of the South African bowlers.
The day had started badly for England with Paul Collingwood dislocating a finger during fielding practice.
The all-rounder went for an x-ray, which showed no fracture, and Collingwood was cleared to continue playing in the test.
The first test was drawn.
Scoreboard
South Africa first innings 343 (G. Smith 75, J. Kallis 75)
England first innings 575-9 declared (A. Strauss 54, A. Cook 118, P. Collingwood 91, I. Bell 141, M. Prior 60)
South Africa second innings
A. Prince c Bell
b Swann 16
G. Smith lbw b Swann 22
H. Amla b Swann 6
J. Kallis b Broad 3
A. de Villiers lbw b Broad 2
J-P. Duminy b Broad 0
M. Boucher not out 20
M. Morkel not out 7
Extras: 0
Total: (six wickets; 32 overs) 76
Fall of wickets: 1-27 2-37 3-40 4-44 5-44 6-50
Bowling: Anderson 7-1-24-0, Onions 4-1-12-0, Swann 12-3-22-3, Broad 9-2-18-3.
