South Africa were 351 for five at the close, an overall lead of 432 on a ground where no team has successfully chased more than 271 in the fourth innings.
In contrast to the first two days, when seam bowlers held sway, South Africa scored freely against a lacklustre bowling attack on a pitch which seemed to have flattened out.
Cook made 117 to lead a powerful batting performance by the hosts.
He shared century partnerships with Dean Elgar (52) and Hashim Amla (48). Cook and Elgar put on 116 for the first wicket, their second three-figure opening stand of the match. Then Cook and Amla added 105 for the second wicket off only 118 balls as the Sri Lankan bowlers and fielders wilted.
Cook, who made his third century in only seven Tests, survived a chance to gully off Suranga Lakmal when he was on 62, but played with more freedom than in previous appearances. He reached his fifty off 81 balls and his hundred off 152 deliveries.
He was out soon after tea when he edged an attempted cut against Dushmantha Chameera to be caught behind. His Test-best 117 was scored off 178 balls with 11 fours.
An indication of the lack of control exerted by the Sri Lankan bowlers was that they only bowled three maiden overs during the innings, two of them by Lakmal, who added the wicket of Elgar to the five he collected in the first innings. But Lakmal, who bowled 27 overs in the first innings, seldom threatened today.
Philander struck twice in the first over of the day after Sri Lanka resumed at 181 for seven. His first ball was a perfect out-swinger which De Silva edged to wicketkeeper De Kock. De Silva had batted impressively to make 43 on Tuesday but Philander made the ball deviate just enough to catch the edge as De Silva played back defensively.
Lakmal hit a boundary but then chipped the final ball of the over to Kyle Abbott at mid-on.
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