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Pakistan were bowled out for 235 in their second innings, an improvement on their first innings total of 156.
But the margin of victory and the series result was a fair reflection of the dominance of South Africa, the world champions, playing in their home conditions.
Dale Steyn took four for 80 and was responsible for the run-out of Azhar Ali which sparked a collapse of the Pakistan middle order.
For the only time in the series, the Pakistan lower order showed some resistance and Steyn was frustrated in a final spell of eight overs after tea as the last pair, Rahat Ali and Mohammad Irfan, played and missed or edged the ball out of reach of the fielders.
At the other end Kyle Abbott was unable to complete a ten-wicket haul on his Test debut, finishing with two for 39 after taking seven for 29 in the first innings.
Finally, captain Graeme Smith turned to left-arm spinner Robin Peterson who trapped Rahat leg before wicket with his sixth ball.
South Africa won the first Test in Johannesburg by 211 runs and the second in Cape Town by four wickets. None of the matches went to a fifth day.
An early finish seemed in prospect when Pakistan lost four wickets quickly after lunch to be floundering at 114 for six, but Sarfraz Ahmed (40) and Saeed Ajmal (31) batted enterprisingly and with a modicum of luck to put on 69 for the seventh wicket.
Azhar Ali and Imran Farhat defied the South African bowlers for most of a morning during which only the wicket of Younis Khan fell. They saw Pakistan through to lunch at 87 for two, with Azhar on 27, eked out over 110 balls.
But Azhar did not face another ball before he was run out by a superb throw from fine leg by Dale Steyn after being sent back by Farhat, who had initially been seeking a second run.
Pakistan's hopes of avoiding an innings defeat crumbled as three more wickets fell in quick succession.
Farhat was caught behind, slashing at first innings bowling hero Kyle Abbott for 43 and without addition to the score Pakistan captain Misbah-ul-Haq edged an away-swinger from Rory Kleinveldt to wicketkeeper AB de Villiers.
Asad Shafiq made only six before he drove Kleinveldt to mid-off. But Sarfraz and Ajmal showed spirit in batting through until tea.