Pakistan were 64-3 at the close after double centurion Ryan Rickelton led South Africa to a first innings total of 615.
Kagiso Rabada struck twice and Marco Jansen once when Pakistan started their innings. The tourists were effectively four down because opening batsman Saim Ayub was ruled out of the match with a broken right ankle.
Babar Azam, forced to open the batting in Ayub's absence, was 31 not out at the close with his side still 551 runs behind.
Left-handed opening batsman Rickelton made a chanceless 259, South Africa's joint seventh highest Test score, before he was seventh man out with the total on 557.
South African captain Temba Bavuma opted to keep Pakistan in the field until the last man, 18-year-old debutant Kwena Maphaka, was out 40 minutes after tea.
By then, South Africa had added 299 runs to their overnight 316-4, scoring at almost five runs an over.
Resuming on 176, Rickelton was content to play the anchor role while Kyle Verreynne hit 100 off 147 balls in a sixth-wicket partnership of 148 off 222 deliveries.
Verreynne holed out to deep midwicket after an innings which included five sixes as well as nine fours, but Rickelton continued to bat flawlessly while Jansen thrashed a 42-ball half-century.
A rapid seventh-wicket stand of 86 was ended when Rickelton was caught at mid-on going for a big hit after batting for 607 minutes. He faced 343 balls and hit 29 fours and three sixes.
Jansen was out for 62 off 54 balls, but Keshav Maharaj hit 40 off 35 deliveries as the bowlers continued to take a pounding.
South Africa's innings ended when Maphaka was bowled second ball for nought to become the 100th Test wicket for Mohammad Abbas.
At 18 years 270 days, Maphaka became South Africa's youngest Test player when the match started on Friday.
Abbas took three for 94, while wicketkeeper Mohammad Rizwan held six catches in the South African innings.
Pakistan captain Shan Masood was caught at first slip off Rabada for two in the first over of his team's reply.
Kamran Ghulam was bowled by Jansen for 12 and Saud Shakeel was out for nought, providing a second catch for David Bedingham at first slip off Rabada.
Pakistan were floundering at 20-3, before Babar and Rizwan put on an unbeaten 44 for the fourth wicket as the day's play ended.
