Fullback Jesse Mogg backed up his brace from the opening weekend with a brilliant individual try in the first half and it was his break that set up a first try since his comeback from retirement for former Wallabies winger Clyde Rathbone.
Australia flanker Scott Higginbotham grabbed his first try for the Rebels, however, and two penalties and a conversion from James O'Connor gave the hosts a 13-12 lead at halftime.
With openside David Pocock leading the Brumbies' dominance of the breakdown, the visitors kept the Rebels scoreless after the break and racked up 18 unanswered points, including tries from skipper Ben Mowen and replacement back Robbie Coleman.
"We knew what we had to do, we just had to flick that switch," said Mowen in a pitchside interview. "We're confident in our systems but things don't just happen, you have to rock up and make them happen."
Mogg has started the new season on fire and only five minutes had gone when he exploded up the sideline, chipped the ball past the last defender and beat Nick Phipps in a footrace to ground the ball.
Some 12 minutes later, Mogg's opposite number O'Connor showed his class by skipping past four defenders before offloading to Higginbotham, who raced across the line to score.
Rathbone, who won 26 caps for Australia before retiring because of injury in 2009, benefited from another burst up the middle from Mogg to touch down in the corner four minutes before the break.
The South African-born winger showed he had lost none of his hard edge when he clattered into O'Connor eight minutes after halftime and forced the Wallabies back off the pitch for the remainder of the match.
With flyhalf Kurtley Beale also struggling with an injured arm, the Rebels were blunted as an attacking force and two Christian Lealiifano penalties and a rampaging run for a try from number eight Mowen took the Brumbies out to a healthy lead.
When replacement back row forward Jarrod Saffy was sin-binned with 11 minutes remaining, the fourth try for a bonus point looked inevitable and Coleman duly delivered by running straight through the hapless Beale.
"They worked themselves back into the game in the second half and we just didn't get enough ball," Rebels skipper Gareth Delve said. "They just strangled us out of it and took their chances."