Oosthuizen, who had missed the cut in all his three previous Opens, never looked like being headed after starting the day four clear and finished with a 71 for a 16-under total of 272.
The 27-year-old sank a short par putt at the 18th, took off his cap to acknowledge the thunderous applause from the crowd, hugged his caddie before embracing his wife and young daughter Jana.
"It's unbelievable, after the 12th hole it actually became difficult having such a big lead, to keep calm, and I'm glad I had those (spare) shots on 18 because I cramped a little with the putter," Oosthuizen told the BBC.
"It's just amazing," he added.
England's Lee Westwood emerged from the pack to finish second with a two under 70 putting him on 279.
Paul Casey, who began the day as Oosthuizen's closest challenger, ended in a tie for third place on eight under alongside Sweden's Henrik Stenson and Northern Ireland's Rory McIlroy.
Oosthuizen became the sixth South African major winner and the first to lift the Claret Jug since Ernie Els in 2002.
Tiger Woods, with the putter he ditched at the start of the week returned to his bag, could not add to his three Claret Jugs after an outward 37 scarred by two double bogeys saw him sink down the field to finish on three-under in a tie for 23rd.
South Korea's Jin Jeong made sure of the Silver Medal for leading amateur, signing off in style with an eagle two at the 72nd hole and a share of 14th place.
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