Sibley, playing in only his fourth Test, came in during the morning session after James Anderson had picked up the last two South African wickets to finish with 5-40 and give England a first innings lead of 46.
By the close, Sibley was still there and England had reached 218 for four, stretching their overall lead to 264.
Sibley’s 85, his first Test half-century, came off a painstaking 222 balls in just under six hours and included 13 boundaries.
It was slow going but it consolidated England’s advantage after Anderson and Ben Stokes combined to secure an important first innings lead.
Anderson completed his 28th Test five-wicket haul – an England record – and Stokes caught last man Anrich Nortje to set an England Test record of five catches by a fielder.
Zak Crawley made a streaky 25 when England started their second innings, with Sibley making only three in an opening stand of 28, before the day settled into a battle of attrition.
Sibley and Joe Denly (31) used up 33 overs in adding 73 for the second wicket before Denly hooked Nortje to fine leg.
England captain Joe Root was more aggressive as he and Sibley added 116 for the third wicket.
Root made 61 off 98 balls before being caught at slip off Dwaine Pretorius shortly before the close.
The ball that got Root was an exception in deviating from a crack outside off stump that was far more of a factor on the first two days.
Dom Bess, who suffered a golden duck in the first innings, came in as nightwatchman and survived until the last ball of the day when he was adjudged to have gloved a bouncer from Nortje to wicketkeeper Quinton de Kock, thus collecting a ‘pair’.
Those two late wickets marked the high point of the day for the toiling South Africans who will be wary of the potential damage that Stokes and Jos Buttler could inflict on Monday.
On a cool, overcast day, South Africa’s bowlers were unable to gain any swing, either conventional or reverse, while the application of the heavy roller before play and after the South African innings seemed to have removed some of the life the pitch had shown on the first two days.
The 24-year-old Sibley came into the match having scored only 71 runs at an average of 14.20 in his first three Tests but the tall, strongly-built opener showed the powers of concentration and patience that made him a heavy scorer for Warwickshire in the 2019 county championship.
Despite the late wickets, England, needing a win to level the four-match World Test Championship, played themselves into a commanding position with two days remaining.
South Africa will be able to claim a second new ball after one over on Monday and will look to strike early if they are to find their way back into the match.
Anderson took both South Africa’s remaining wickets as the hosts added only eight runs to their overnight total.
Stokes held his fifth catch of the innings, all taken at second slip to dismiss last batsman Nortje off Anderson’s bowling.
It was the first time in 1,020 Test matches that an England fielder held five catches. There had been 23 instances of four catches in an innings, most recently by England captain Joe Root against Ireland at Lord’s last year.
Stokes equalled the world Test record of five catches, which had been achieved 11 times previously, most recently by Steve Smith of Australia in 2017-18, also at Newlands.
Anderson moved one ahead of Ian Botham on the list of England bowlers with most five-wicket hauls.