The Redbacks are 4-382 at stumps on Friday's second day of play, a commanding 151-run lead over the Bulls.
Cooper hammered his 11th first-class century after being a last-minute inclusion for the fixture.
The 31-year-old had been dropped but was recalled when teammate Callum Ferguson was ruled out before the opening day's play because of a calf injury.
Cooper cashed in on his teammate's misfortune, smacking 22 fours and two sixes in his aggressive 246-ball knock.
Cooper was superbly supported by Jake Lehmann, who is unbeaten on 76, and opener Jake Weatherald, who made 69.
Cooper figured in a telling 107-run partnership with the highly-rated Weatherald, who was in good touch but failed to convert his promising start into a big hundred which could have raised his Test selection reckoning.
The lefthanded Weatherald fell when nicking an attempted drive outside off stump from the bowling of Luke Feldman (1-55).
SA captain Travis Head put on 62 runs in a stand dominated by Cooper before he was caught behind for 16 from a Michael Neser (1-49) delivery.
Lehmann then joined Cooper and the pair scored freely in a 143-run union against a sagging Bulls bowling attack.
Lehmann struck eight fours and a six in his 111-ball innings as the Redbacks took a stranglehold on the game, with Cooper dismissed late in the day when edging a Brendan Doggett short-ball to the wicketkeeper.
Doggett was the only Bulls bowler to claim multiple wickets, taking 2-62 from 23 overs - he also dismissed opener Conor McInerney for 24.