Left-arm spinner Noman Ali completed a 10-wicket haul and seamer Shaheen Shah Afridi took the last three wickets as world champions South Africa, who were 2-51 overnight, were dismissed for 183 in the middle session on Wednesday.
South Africa were always going to be up against it when Tony de Zorzi, who scored a century in the first innings, was trapped leg before wicket by Shaheen Shah Afridi in the first over of the morning, as they chased a target 277 to extend a 10-Test winning streak.
It was a rare wicket for a seamer in a contest where neither side used them much, such was the spin-friendly condition of the turning wicket.
Tristan Stubbs lasted only eight balls before a viciously turning delivery from Noman deceived him, bringing Brevis to the wicket.
The 22-year-old already has a reputation in limited overs cricket for his expansive hitting and did not temper his approach in his third Test appearance.
He made up for his first ball dismissal in the first innings with 54 runs off as many balls as he put the pressure back onto the Pakistan bowlers and offered South Africa the brief potential of an unlikely win.
Together with Ryan Rickelton they put on a 73-run partnership, but Noman had the last laugh with a turning delivery that clean bowled Brevis.
Rickelton then got a leading edge to Sajid's turning ball to depart for 45, leaving the tourists 6-137 at lunch.
The last four wickets fell without much resistance with Afridi given another spell to wrap up the tail and return figures of 4-33. Noman took 4-79 to add to his six wickets in the first innings.
"We built enough margins where we could have the upper hand and I'm glad we finished it off," said Pakistan captain Shan Masood.
Defeat for South Africa marked a losing start to the defence of their World Test Championship crown at the beginning of a new two-year cycle.
"There was some good stuff over the few days that we played, I thought, really good signs, I suppose, bat and ball," said Proteas skipper Aiden Markram.
"So we have a bit to take out, bit to learn as well, and clean up our space for the next game."
Pakistan made 378 and 167 while South Africa replied with 269 and 183.
Next week's second Test will be played in Rawalpindi.
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