Set a near-impossible 491 to win, South Africa successfully batted through the final day to finish on 287 for five after resuming on 85-2 then losing two early wickets.
Rudolph, who was called into the team as a late replacement for the injured Jacques Kallis, survived for more than seven hours to finish unbeaten on 102, frustrating the Australian bowlers with his sturdy defence and unwavering patience.
The Australians looked to be on course to win the match when they reduced the visitors to 138-4 before lunch but managed only one more wicket.
Rudolph and Justin Kemp held the fort for nearly three and a half hours, putting on 112 runs for the fifth wicket until Shane Warne dismissed Kemp for 55 after tea, leaving Mark Boucher (13 not out) to play out the last hour with Rudolph.
Rudolph, who started the day on 18 and was exhausted by the end of it, batted 431 minutes and faced 283 balls. He raised his fifth century just five overs before the close when he drove Warne through the covers for his 13th and final boundary.
South Africa made a terrible start to the day when Australian fast bowler Brett Lee dismissed Herschelle Gibbs and Warne removed Ashwell Prince lbw before lunch.
Gibbs fell for 33 when Lee found a thick edge and Warne took a sharp, low catch at first slip.
Warne, pitching the ball into the footmarks to try and extract more turn from a flat wicket, bowled a marathon 47 overs in the second innings, taking 3-83, but could not find a way behind the South Africans' defence.
With no real hope of winning, the tourists had no alternative other than to batten down the hatches with just 202 runs scored on the day, including 50 between lunch and tea when the result was still in the balance.
Rudolph crawled to his half-century off 150 balls in 230 minutes but did not offer the Australians a single chance, undermining their decision not to declare the previous day.
Kemp had a lucky escape in the last over before tea when he almost ran himself out but Glenn McGrath's throw missed the stumps.
He reached his half-century off 143 balls in a little over three hours but his resistance gave way on 55 when he was caught by Ricky Ponting at bat pad off Warne.
The second Test starts in Melbourne next Monday with the final match in Sydney from January 2.
Australia 1st innings 258
South Africa 1st innings 296
Australia 2nd innings 528 for 8 decl
South Africa 2nd innings (Overnight: 85-2; Target: 491 runs)
G. Smith lbw b Bracken 30
A. de Villiers c Hodge b Warne 12
H. Gibbs c Warne b Lee 33
J. Rudolph not out 102
A. Prince lbw b Warne 8
J. Kemp c Ponting b Warne 55
M. Boucher not out 13
Extras (b-18 lb-13 nb-3) 34
Total (for 5 wickets, 126 overs) 287
Fall of wickets: 1-35 2-55 3-109 4-138 5-250
Did not bat: S. Pollock, C. Langeveldt, A. Nel, M. Ntini
Bowling: G. McGrath 24 - 11 - 39 - 0 (nb-1), B. Lee 31 - 9 - 83 - 1 (nb-2), S. Warne 47 - 21 - 83 - 3, N. Bracken 19 - 5 - 37 - 1, A. Symonds 3 - 0 - 6 - 0, B. Hodge 2 - 0 - 8 - 0