Chasing an improbable 482 to win after Australia declared at 4-290, the Proteas never looked likely after Johnson ripped through their top order.
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He dismissed openers Graeme Smith (4) and Alviro Petersen (1) and finished with second innings figures of 5-59 off 16 overs.
Much like England in the Ashes, South Africa had no answers for Johnson's hostility and raw pace and his match figures of 12-127 saw him become the first Aussie to take 12 scalps in a Test since Jason Krejza achieved the feat on debut against India in 2008.
Michael Clarke's side take a 1-0 lead in the three-match series, while for the hosts, it represents their heaviest defeat in a Test since 2010, and their biggest loss at home in over four years.
The tourists began day four at 3-288 and had only added two when Shaun Marsh was caught behind off Dale Steyn (2-61 off 14.2 overs) without adding to his overnight score of 44.
That dismissal prompted a declaration from Clarke and South Africa's run chase immediately hit a setback as Johnson removed both openers in the first four overs.
Petersen edged a delivery to wicket-keeper Brad Haddin and in the fourth over Smith followed him back to the pavilion after debutant Alex Doolan took a great catch at short leg.
Faf du Plessis (18) and Hashim Amla (35) began to build a third-wicket union for the hosts but as he so often does, Peter Siddle (2-55 off 16) broke a key partnership for Australia.
Du Plessis was trapped lbw to Siddle and Amla, who sit six fours in his 35, soon followed when he edged Ryan Harris (2-35 off 12.4) to first slip.
Johnson then took over as he removed JP Duminy (10), Ryan McLaren (6) and AB de Villiers (48) as South Africa collapsed from 4-128 to 7-151.
Australia did not take their foot off the pedal as Siddle bowled Robin Peterson (21) and Harris removed Steyn (3) before victory was secured when Morne Morkel (1) ran himself out in the 60th over of South Africa's never-likely chase.
The victory is Australia's sixth consecutive Test win, and their first away from home since April 2012 against the West Indies.