With so much time left in the test, a declaration at that point would make South Africa heavy favourites to win or secure a draw at the very least.
Later on, during the drinks break, Beakley and team doctor Richard Saw again assessed Paine's sore thumb.
Australia are still in trouble as they still trail by 287 runs on the first innings, but have slowed the victory charge of the home side who seek a first home test series win over the tourists in nearly 50 years.
Elgar will resume on the fourth morning, with his captain, Faf du Plessis, on 34, as the pair look to increase the lead towards 500 runs.
Aiden Markram contributed 37, in the process becoming the second fastest South African to 1000 Test runs, while opening partner Dean Elgar hit a patient 39 not out from 158 deliveries.
Australia need victory to avoid a first series defeat in South Africa since 1970, but have been rocked by the ball-tampering scandal that resulted in former captain Steve Smith, his deputy David Warner and batsman Cameron Bancroft being sent home and given hefty bans after the third test in Cape Town. Thus, prior to the start of play and just after the completion of the national anthems of both sides, Tim Paine and the rest of his Australian side went ahead and walked past the Proteas in a bid to shake their hands, something that he was lauded for and depicting a hallmark of his sportsmanship.
Tim Paine had suffered a hairline fracture on his right thumb on Saturday, the Australian confirmed on the eve of the start of Day 3 at New Wanderers.
Paine continued to keep in Johannesburg then came out to bat later in the day when Australia collapsed to 5-96.
Paine was last man out to a running, diving catch by Elgar, which gave the innings a gripping finish.
Sayers was one of four changes for Australia, three of them directly forced by the fallout from the tampering affair.
"We've got to try and take some sort of positive out of it long term and turn ourselves into the cricket team we want to be". South Africa wasn't yet sure if he'd be able to bowl during the last two days of his test career. The third test played between the two teams witnessed one of the most controversial ball-tampering cases in the recent history of cricket.