Hawk-Eye ball-tracking technology, as carried by ESPN Cricinfo, indicates Anderson had the lowest average speeds of all the specialist seamers in the third Test – coming in at 83.35mph across two innings.
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But Temba Bavuma (32no) and Quinton de Kock (25no) seized back the initiative with a lively late stand.
Amla’s 25th test century came off 131 balls in a confident display which brought him 18 fours while Cook was nine runs short of a debut test century at Centurion Park on Friday.
He also carried his bat against this England attack in the tour match, but he has often been overlooked for Test duties.
Elgar had advanced down the track before making good contact as he looked to hit one through mid-wicket, but Taylor showed superb anticipation to throw himself in the way of the ball before it nestled in between his thighs as he claimed a quite freakish catch. Cook partnered with Hashim Amla for a 202-run stand, South Africa’s highest since Amla and AB de Villiers put on over 300 at this very same venue against the West Indies back in 2014.
Cook, the son of former South Africa test player Jimmy Cook, has meanwhile been the country’s in-form opening batsman in domestic cricket, leading to repeated calls for his inclusion.
Moeen Ali then tied things up as far as England were concerned, trapping JP Duminy lbw, the spinner’s second victim after he had induced Taylor’s spectacular catch – held between his legs – with his fourth ball of the day.
“I’m really looking forward to this week to see how people respond”, said Cook. The 150 partnership came up from just 204 balls, and despite England’s best attempts to stop him with an eight-one field and an offside line, Amla’s hundred came up from just 131 deliveries shortly before tea.
Cook, who had played 155 first-class matches before being selected, added: “As time went on I knew it was getting less likely that I would make my debut but I never gave up hope”.
Broad, who has been eager to return to England’s limited-overs side since the summer, will feature in the five-match one-day series but has not been included for the two Twenty20s against the Proteas.
Cook had a life on 47 when Bairstow again dived in front of his captain Cook and spilled a hard chance off Broad. Quinton de Kock, Kyle Abbott and Dane Piedt were also selected for the game.
England have already won the series with an unassailable 2-0 lead, but they are now on the backfoot as the Proteas reached stumps on 329-5.
Chris Woakes, replacing the injured Steven Finn, was the worst offender, serving up 13 chanceless overs for 65.
The one wicket to fall before lunch was thanks to another brilliant piece of fielding at short-leg by James Taylor to remove Dean Elgar.
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JP Duminy, controversially recalled to the team in place of Faf du Plessis and playing for his Test career, looked solid as he went to 16 and added 33 for the fourth wicket with Cook.
