South Africa were bowled out for 387 in their second innings at stumps on day four of the series-deciding Test on Friday, following a late collapse that put the game back in the balance.
After losing just one wicket through the first five hours of play and building their score to 5-335 with a lead of 235, the Proteas lost 5-52 before the close on Friday.
Still, the hosts will feel they have given themselves a strong chance at winning the third Test, after starting Friday at 4-140 and with little hope of victory.
Mitchell Starc took 3-65 for Australia to make it nine for the match, as Rickelton and Mulder's excellent centuries led the resistance for South Africa after they shared in a 154-run sixth-wicket stand.
But when Josh Inglis did superbly well to cut off a Mulder reverse sweep at backward point and ran Rickelton out for 101 at the bowler's end, Australia were alive again.
Starc had a hooking Marco Jansen out for a duck, before Simon Harmer also went for zero when a Nathan Lyon ball bounced off his bat handle and onto the stumps as he tried to leave the delivery.
Marnus Labuschagne then ran out Gerald Coetzee in another mix-up, before Mulder (115) holed out in the deep off Pat Cummins (2-71) in the final over of the day to end the innings.
Australia's task is a tough one as the largest fourth-innings target successfully chased at Newlands was 331 by Steve Waugh's side in 2002, with the next best being 236 by South Africa in 2011.
There had been a thought on day three that anything beyond 150 could be tricky to chase, with plenty of spin on offer.
But while the ball still turned a long way on Friday, a lack of pace and bounce in the wicket made it far less threatening for batting even if run-scoring was not easy.
Australia's tiring quicks struggled to get any real purchase out of the Newlands pitch, with short balls often bouncing before reaching wicketkeeper Alex Carey.
At one point, Josh Hazlewood could be heard saying over an effects microphone: "Good luck bowling bouncers on that thing."
Whether the Proteas are able to get more out of the wicket on day five will likely prove crucial in this contest.
The Proteas were guilty of throwing away their only wicket in the opening two sessions, with Matthew Breetzke (43) the fifth South African out hooking this Test.
Breetzke attempted to take on a shorter ball from Pat Cummins and swiped the air with his bat in frustration, knowing he had let an opportunity go.
Captain Cummins had endured the longest wait for a wicket in his Test career until that point, having bowled 374 balls and conceded 210 runs since his last scalp in Durban.
As well as the tantalising outcome to the series, the first place in the ICC rankings is on the line as well as a top-two spot in the Test Championship standings.
The Proteas also have an unbeaten run in home Test series, dating back to early 2020, to protect.
Fightbacks have been the theme of this series, with the heavyweight bout between the world's two top-ranked teams set for the conclusion it deserves.
Australia were 6-124 in the first Test in Durban before Matt Renshaw's career-defining 190 hit them to 390 and a big win to start the series.
South Africa were then 6-110 in Gqeberha last week, before Temba Bavuma's century took them to 9-425 declared and Australia were flogged after being bowled out for 99 in reply.
Even in this Test, the Proteas reached 0-140 at lunch on the first day, before Starc's six-wicket haul put Australia back in the game and the tourists eventually claimed a 100-run first-innings lead.
