England were 179 for four at the close of play after being sent in to bat. Fast bowler Dale Steyn spoiled a good recovery by the tourists when he struck shortly before the close, having Taylor caught behind for 70.
Compton and Taylor came together with England struggling on 49 for three and forged a partnership of 125 off 274 balls.
Taylor was dismissed in fading light, capping a fine comeback from injury by Steyn, who took three for 29.
Compton and Taylor batted solidly on a slow pitch which did not offer the bowlers as much assistance as South African captain Hashim Amla might have hoped when he won the toss and sent England in to bat in overcast conditions.
Compton, returning to the city of his birth and early upbringing, was resolute and almost strokeless at times as he made 63 not out off 179 balls, with four boundaries.
After rain delayed the start, only 27 minutes of cricket was possible before rain brought an early lunch, while there was a 22-minute rain break during an extended afternoon's play.
Steyn, who had not played any cricket since suffering a groin injury on the first day of a series in India on November 5, found his rhythm almost immediately and had skipper Cook caught at second slip by Elgar for a duck off the fifth ball of his second over. He did not concede a run in the 3.1 overs he bowled before lunch.
When play resumed, he faced only four more balls without adding to his score of ten before edging a Steyn outswinger to wicketkeeper AB de Villiers.
Root looked busy at the crease and made 24 off 37 balls, including a pulled six off Abbott. But he went back on his stumps when Piedt came into the attack and was trapped leg before wicket by a ball which spun back at him.
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