The home side scored nine tries, while fly half Matt Giteau slotted six conversions for 12 points before he was replaced with 20 minutes remaining.
The Brumbies move into fifth place in the table with 26 points, two behind the fourth-placed Stormers. Fullback Julian Huxley added two conversions after Giteau went off.
"Happy with the amount of tries we scored ... but we butchered quite a few opportunities and probably should have had more points in the first half," Brumbies captain Stephen Hoiles said in a televised interview.
"It puts us in a good position but let's be honest we are facing some pretty tough teams from now on, so it will get pretty serious from now on in." Cheetah’s fly half Naas Olivier opened the scoring with an early penalty but it was one of the few times the visitors got inside the Brumbies territory in the first half.
The Brumbies replied with three first-half tries to Hoiles, scrumhalf Patrick Phibbs and center Tyrone Smith. Giteau converted all three tries to give the Brumbies a 21-3 lead at the break.
Wallabies captain Rocky Elsom secured the bonus point for the Brumbies early in the second half before Cheetahs replacement scrumhalf Sarel Pretorius darted over.
Olivier, however, missed the easy conversion and was immediately replaced by Riaan Viljoen.
Loosehead prop Ben Alexander and center Adam Ashley-Cooper both scored tries for the Brumbies before Cheetahs hooker Adriaan Strauss rumbled over and Viljoen converted.
Matt Toomua scored a late try for the Brumbies then injured himself when he collided with the padding around the goalpost, before Alexander crashed over for his second try of the match and Alfi Mafi crossed in the final minute.
"There is not a lot to be said, we were very disappointing and we have a lot of hard work to do," Strauss said.
