The Proteas had won each of the first two Tests with a day to spare and went one better at Centurion, wrapping up the tourists' second innings for 235 on the third evening.
Dale Steyn took four for 80 in the second innings while Kyle Abbott took nine wickets on his Test debut.
Half-century partnerships between Azhar Ali and Imran Farhat and the unlikely pairing of Saeed Ajmal and wicketkeeper Sarfraz Ahmed were in vain as Pakistan managed only a minor improvement on their first-innings capitulation to 156 all out.
The visitors began the third day on 14 for one, with Azhar and Younus Khan at the crease, having followed on 253 runs behind.
However, the partnership was broken on 38 by Steyn, with Younus edging to captain Graeme Smith at first slip on 11.
Azhar and Farhat patiently negotiated the remaining overs before lunch but after their partnership reached 54, a mix-up as they turned for a second run saw Azhar sent back by his partner and run out for 27 as Steyn fizzed in a powerful throw and AB de Villiers demolished the stumps.
Farhat followed for 43, slashing a short ball from Kyle Abbott to De Villiers, as the rot set in.
Misbah-ul-Haq (five) offered another easy catch to the wicketkeeper with a thin edge off Rory Kleinveldt and Asad Shafiq (six) then popped a leading edge off the same bowler straight to Vernon Philander at wide mid-off, with Pakistan still 139 behind.
Two reviews then went in the batsmen's favour, Ajmal successfully overturning an lbw decision as Hawk-Eye showed Robin Peterson's delivery was bouncing over the top, before South Africa failed to have Sarfraz caught at slip.
Ajmal paddled Peterson neatly over De Villiers' head for two and lofted the next ball down the ground for six as the pair offered some overdue resistance with a stand of 62 by tea.
The interval arrived with Sarfraz unbeaten on 30 and Ajmal on 27, but the latter added only an additional boundary in taking the partnership to 69 before being dismissed lbw by Steyn, with DRS this time unable to save him.
Again, the departure of one set batsman was swiftly followed by another. Sarfraz had advanced to 40 from 45 balls before slashing Steyn to third man, where Dean Elgar took a very well judged catch.
Ehsan Adil hit two boundaries before becoming Abbott's ninth victim of the match, caught at mid-on for 12.
Rahat Ali hit five boundaries, two of them edged off Steyn in an over that also saw him successfully overturn a bat-pad decision against him, but was trapped lbw by Peterson to end the match.
Abbott's match figures of nine for 68 were the third best ever by a Proteas debutant, bettered only by Alf Hall and Sydney Burke who took 11 wickets apiece.
The two teams will now contest two Twenty20 internationals, next Friday and Sunday, followed by five one-day internationals.