A penalty goal to Stirling Mortlock after a ruck infringement opened the scoring for the Brumbies after five minutes.
From that point on, the Reds dominated the first half, winning nearly ever scrum and benefiting from poor throwing into the Brumbies lineout from veteran hooker Jeremy Paul, who was replaced at the interval.
Reds winger Brando Va'alu looked like he had scored in the corner in the 23rd minute following a break up the middle by outside centre Peter Hynes.
But the referee pulled play back for a forward pass from fly-half Berrick Barnes from the ruck, and the Reds instead had to settle for a penalty goal to full-back Clinton Schifcofske after advantage had been played for the Brumbies being offside.
The Reds went close again just 10 minutes later, but solid Brumbies defence and dropped ball ruined the Reds build-up with the score remaining 3-3 at half-time.
In contrast to the first half, the Brumbies came out firing in the second half but also suffered from uncharacteristic dropped ball as the referee's whistle came more into the game for ruck and scrum infringements.
The game breaker came in the 71st minute when Mortlock, who has just returned to the Brumbies line-up after suffering concussion from the first round match against the Chiefs, was coathangered by a high tackle from Lloyd Johansson.
Johansson was sin binned for the high hit, Mortlock was assisted off the field and Brumbies full-back Julian Huxley slotted over the conversion from almost directly in front of the posts to make the score 6-3.
A 14-man Reds outfit tried for a late score but the Brumbies defence held firm to register the win.