Set a huge victory target of 539, Australia were dismissed in their second innings for 361 on the final day as they succumbed to their fourth straight Test defeat.
Australia hadn't lost the first Test of a home season since 1988, but Rabada stepped up after Dale Steyn's shoulder injury to help South Africa complete a remarkable victory.
Debutant spinner Keshav Maharaj (1-94) delivered the final blow, trapping number 11 Nathan Lyon lbw for eight, with Peter Nevill unbeaten on 60.
Steven Smith's Australia lost all three Tests on their tour of Sri Lanka in July and August, and they will now look for a quick recovery at this week's second Test in Hobart.
Resuming at 169 for four, a draw was Australia's only hope and it appeared a forlorn task, with man-of-the-match Rabada ensuring that was the case with 5-92.
With star paceman Steyn suffering a shoulder injury that will sideline him for at least six months, Rabada stepped up in fine style to lead the South African attack.
The 21-year-old claimed three wickets on day four and two more today to record his fourth five-wicket haul in just nine Tests, a record for a South African player of his age.
The rot set in early for Australia on the final day, when a searing Rabada yorker hit Mitchell Marsh on the pad on 26, although he was given not out by umpire Aleem Dar.
The South Africans reviewed the decision and were successful, ending a 50-run stand between Marsh and Usman Khawaja and striking the first blow of the day.
Rabada then grabbed his fifth wicket when he trapped Mitchell Starc lbw with another fast yorker for 13.
Rabada is shaping as a perfect long-term replacement for Steyn, having also claimed the wickets of Shaun Marsh (15), Smith (34) and Adam Voges (1) on the fourth day.
JP Duminy capped his fine match, having made a century in South Africa's second innings, by claiming the wicket of Khawaja with his part-time spin, lbw for 97.
Dimunitive batsman Temba Bavuma, who produced one of the great run-outs on the fourth day to dismiss David Warner, the first Australian wicket, was brought on to bowl and almost struck with his first ball in Test cricket.
It skidded through and struck Khawaja right in front of the stumps, but Bavuma had overstepped and it was a no-ball, denying him the honour of becoming just the 21st bowler to claim a wicket with their first ball in Test cricket.
However, Bavuma had his moment later in the innings, breaking an obdurate 65-run wicket ninth-wicket stand by having Josh Hazlewood caught off a leading edge for 29.
South Africa, who looked almost out of the match on the second day when Australia cruised to 158 for none in replay to the Proteas' 242, declared their second innings on 540 for 8 after lunch on the fourth day to set Australia a record run chase.