Travis Head was dismissed by Nortje early on Day 3 soon after completing his half-century.
Anrich Nortje made it two from two by bowling the returning David Warner for 200.
Nathan Lyon scored a brisk 17-ball 25 as Australia ticked their lead past 250.
Alex Carey and Cameron Green put on 117 for the 8th wicket to propel the lead past 300.
Alex Carey scored his first Test century and became only the second Australian keeper-batter, after Rod Marsh, to score a century in the Boxing Day Test.
Cameron Green battled a finger fracture to score a gutsy half-century to go with his first innings five-fer.
Australia declared with an overall first-innings lead of 386
Pat Cummins then dismissed his opposite number, Dean Elgar, for a duck.
Rains bought an early end to the day's play with South Africa still 371 behind.