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Mogg also kicked two penalties while his forwards, bolstered by impressive new signing David Pocock on the openside flank, edged the battle of the breakdown in a scrappy encounter between the only Australian sides to have won the competition.
"We're happy, wins are important at this time of year and they're a side who outplayed us twice last year," Brumbies skipper Ben Mowen said.
"That's okay for a base but where we want to go, that won't be good enough."
Australia got its Super Rugby season underway a week early to allow them a longer break for the British and Irish Lions tour in June and July.
There was little to worry anyone watching from the northern hemisphere, though, in a dour contest dominated by rugged defence and ill-discipline rather than free-flowing backplay.
The Brumbies set the tone by conceding three penalties in the first five minutes with Reds fullback Mike Harris converting just one, albeit from near halfway.
Mogg scored his first try from his team's first attacking ball, beating Wallabies winger Digby Ioane in a foot race to touch down after scrumhalf Nic White's neat grubber kick had bounced kindly.
Harris put the visitors back in front 6-5 after 25 minutes but White, taking over kicking duties from misfiring flyhalf Christian Lealiifano, converted his first penalty to send the Brumbies into halftime with a slender 8-6 advantage.
Mogg's pair of penalties at the start of the second half gave the twice champions a lead they never looked like relinquishing and White slotted his second from distance with six minutes to go.
Quade Cooper, playing on the backfoot for most of the match and without his injured halfback partner Will Genia, made a quiet return for the Reds after a long spell on the sidelines.
Trying to create something from nothing in the final minute, the Wallabies flyhalf floated a pass towards the wing which Mogg snatched out of the air before cantering over the try line untouched to seal the win.
"It was great defence from the Brumbies and there weren't many chances for us," said Cooper. "There were a few, but ill-discipline let us down a little bit."