The tourists started day five knowing that only six wickets separated them from victory over the world's number-one ranked Test side and they did not waste time making their way through the order.
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Ali finished as England's top wicket-taker with seven dismissals, the last of which was removing Kyle Abbott for two lbw.
JP Duminy (26*) provided some late resistance, but tail-enders Dane Piedt (0) and Morne Morkel (8) were soon gone to Chris Woakes and Stuart Broad respectively as England wrapped up victory before lunch on the final day.
South Africa, who suffer their heaviest-ever defeat at Kingsmead, will look to bounce back in the second Test, which begins in Cape Town on Saturday.