Sibley converted his overnight 85 into 133 not out as England piled on the runs in the second Test at Newlands, declaring on 391 for eight with a towering lead of 437.
While Sibley continued playing his anchor role to aplomb, racking up 311 deliveries over more than eight hours at the crease, it was Stokes who reset the tone with a vivacious 72 in just 47 balls.
Play resumed on 218 for four, Sibley understandably watchful with his milestone in sight, but Stokes was intent on destruction at the scene of his career-best 258.
South Africa initially declined to take the second new ball, but if they thought the old one would be harder to hit, they were soon advised otherwise.
A pulled four got Stokes' blood pumping before a glorious blow for six soared back over Dwaine Pretorius' head and a reverse pull underlined the message. Keshav Maharaj was next on the receiving end, heaved for six into the stands.
Vernon Philander promptly took the new ball and shipped 12 runs in five balls to Stokes – the same number he had conceded in his first 13 overs.
Stokes toe-ended one pull off Rabada on 38 but saw the ball squirm through Quinton De Kock's gloves as he raced towards short fine-leg.
Sibley's quiet progress had slipped into the shadows but he reclaimed centre stage as his big moment neared. A thick edge off Rabada took him to 99 but he got over the line in more stylish fashion, stooping to sweep Maharaj for four.
His century had taken 269 deliveries and almost seven hours, the kind of sober knock Test victories are often built on.
Stokes passed fifty in a whirlwind 34 balls and was threatening to catch his partner until the fun stopped, another big blow held at long-on to cease Maharaj's suffering.
England's charge would not be halted, though, with Sibley even entering into the spirit when he crashed the spinner for his first Test six.
England continued for 22 minutes after lunch, losing Sam Curran to the third ball of the session, but Root called his men in with the target 20 clear of Test history's biggest ever chase.
Malan and Elgar both hit their first balls for four as the fourth innings began and the trail stayed cool throughout the afternoon.
An inside edge put to an early review for lbw against Malan and of the six bowlers to try their luck, it was part-time leg-spinner Joe Denly who caused the most uncertainty.