The Reds lived dangerously, however, and the Bulls almost levelled the scores with the last move of the match only for centre JJ Engelbrecht to step on the touchline before he dived over the try line to touch down.
Cooper, back in tandem with halfback partner Will Genia for the second match in a row, had added 13 points with his boot to help the 2011 champions get their campaign back on track after a shock defeat to the Western Force in their last outing.
The three-times champion Bulls, who defended stoutly throughout, were kept in the match by five penalties from young fly-half Louis Fouche and another from replacement Morne Steyn, punishing the frequent errors by the Reds.
"We really shot ourselves in the foot at times but I'm really happy that I got through the 40 minutes and even happier with the win," said Australia captain Horwill, who played only the first half after 10 months out injured.
"You saw what Will and Quade can do with a bit of time and space."
The match was level at 6-6 after 36 minutes when Cooper picked up the ball at the breakdown and set off through a huge gap in the defence before offloading to Jono Lance when he was swallowed up by tacklers as he closed in on the try line.
Lance, who had already spent 10 minutes in the sin bin for a dangerous tackle, made no mistake from close range and the Reds finally got the try their earlier pressure had deserved.
After turning around 13-6 behind, however, the Bulls edged back in front when Fouche converted three penalties in six minutes soon after the break.
Another harsh-looking yellow card for a dangerous tackle sent Bulls winger Lionel Mapoe into the sin bin soon afterwards, though, and four minutes later Cooper was dancing through another gap in the stretched defence to put the Reds ahead again.
Cooper added his second penalty on the hour mark but Steyn responded 10 minutes from time to keep alive the Bulls' hopes of a first win at Lang Park - hopes which were only finally dashed by Englebrecht's stray foot after the final hooter.
"It was a tough game, it's always tough here," said Bulls captain Pierre Spies. "Up until the end it could have gone either way but credit to the Reds, they played really well and were hungry for the win."