The hosts slumped to 127 for six in their second innings at lunch on the third day, a lead of just 23 runs. South Africa had been dismissed for 205 in their first innings.
Australia replied with 308.
Fast bowler Clark, who took five for 55 in the first innings, took another two for 22.
The South Africans struggled against Australia's disciplined bowling and on a pitch that still offered movement off the seam.
The pitch claimed 23 wickets on the first two days of the match, and another three tumbled for the addition of just 57 runs in the morning session.
South Africa's main hope of taking the match into a fourth day appears to rest with the patient Jacques Rudolph, who took 29 balls to get off the mark and faced 83 balls for his 23 not out.
Andrew Hall was 10 not out. South Africa's only remaining recognised batsman was Nicky Boje.
Clark struck with the ninth ball of the day when Jacques Kallis attempted to cut and edged to wicketkeeper Adam Gilchrist to be caught behind for 36.
Ten overs later Ashwell Prince dabbed at a leg-side delivery from Clark and was caught behind for 27.
South Africa were reduced to 108 for six when Mark Boucher (2) tried to play a ball from fast bowler Michael Kasprowicz through the on side and was well held by a diving Justin Langer at midwicket.
Boucher was felled by the second ball he faced, bowled by Clark, and play was held up for several minutes while he received treatment.
Boucher recovered enough to continue his innings, but never looked settled at the crease.