Aside from those two stand-out innings, Cook making an assured 115 on his belated bow and Amla adding a fluent 109 to a 202-run stand, it was a day that lived up to its “dead rubber” billing at Centurion.
England, with the series won in style last week, were flat in the field and uninspired with the ball, while the Proteas finished on 329 for five after a middle-order wobble in the evening session.
Cook’s knock was a personal triumph – at the age of 33 he must have thought his time had gone. He was cheered on by his father Jimmy, who made his own debut in 1992 and bagged a golden duck.
Cook had been dropped on 47, Jonny Bairstow’s troubled series behind the stumps continuing as he grassed a one-handed chance at the start of the afternoon session, and Amla was also reprieved early. He had just five to his name when Ben Stokes found the edge with a lovely outswinger, but Bairstow dived in vain across Alastair Cook, who faltered at slip.
None of England’s five-man attack came close to their best, allowing Amla to dominate as he scored his sixth Test hundred against England and 25th overall. But Chris Woakes would be most disappointed with his work, erring with both line and length as his 16 overs cost 74.
A South Africa side showing five changes from the third Test opted to bat first and reached 107 for one by lunch.
Cook avoided his father’s first-ball fate and settled quickly against an off-colour attack. Indeed, the only breakthrough relied heavily on another James Taylor special at short leg. Although not as aesthetically pleasing as his two wonder catches at the Wanderers, Taylor showed bravery and quick thinking to assist Moeen Ali in removing Dean Elgar.
Standing up to a cleanly-hit shot, he deflected the ball into his hunched body and shuffled awkwardly for a few seconds before locating it with his hand.
Stuart Broad finally persuaded Cook to edge one but, not for the first time in the series, Bairstow’s glovework let him down as he fumbled a one-handed effort.
At the other end, Woakes was struggling badly, sending down a three-over spell that cost 26 runs and saw Amla thump five boundaries.
England delayed Amla his inevitable century by packing the offside and hanging the ball wide outside off stump, but when James Anderson eventually straightened up Amla punched him for two to reach three figures.
Surprisingly, he did not kick on after tea, playing on to Stokes just after bringing up the 200 stand.
The stage seemed set for AB de Villiers but he came and went for nought, Broad dismissing the home captain for the tenth time via a smart take from Joe Root at second slip.
That double strike may have unsettled Cook, who hovered nervously in the nineties – surviving one tight lbw referral – before reaching his century with a nudge off his pads.
His innings eventually ended when he diverted Woakes into his stumps, with JP Duminy following for 16 as Moeen struck again.
But Temba Bavuma (32no) and Quinton de Kock (25no) seized back the initiative with a lively late stand.