Brett Lee hit the winning runs, punching Shaun Pollock square on the off-side for four on Tuesday.
Effectively, Australia's last pair were at the wicket, with Justin Langer not expected to bat after suffering concussion when struck by a ball from Makhaya Ntini Saturday.
Lee's partner at the end was Michael Kasprowicz, the same combination who fell two runs short of England's total after adding 59 runs for the last wicket in Birmingham last year.
Lee and Kasprowicz added an unbeaten 19 Tuesday as Australia completed their first clean sweep in a series in South Africa. It was also the first time since 1895/96 that South Africa lost every match in a home series of three Tests or more.
Resuming at 248 for six, needing 44 more to win, Australia had to negotiate the second new ball, taken nine balls into the day.
Damien Martyn went to a century, nudging Pollock for four to third man, but then was trapped leg before by four balls later, after twice edging Pollock, short of wicketkeeper Mark Boucher and first slip, substitute Andrew Hall.
Martyn was out for 101 when the total was 258.
Stuart Clark hit two boundaries in an over from Ntini but in trying to force another shot he top-edged Ntini and Boucher held the catch.
Lee and Kasprowicz both batted confidently as they neared the target, with Lee hitting Pollock through the covers for four and Kasprowicz doing the same to Ntini before Lee hit the winning runs.
Lee, who made 64 in the first innings, finished with 24 not out.
Ntini took four for 78 and had match figures of ten for 178, his third ten-wicket haul in Tests.
Australia's 3-0 series win followed a 2-0 win in a three-match series in Australia earlier in the season.