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The Brumbies dominated the back-to-back champions to lead 36-9 after 63 minutes with 18 straight points while Chiefs replacement James Lowe was given a yellow card for tripping.
The Chiefs hit back with two late converted tries to reduce the final margin.
Try doubles to man-of-the-match Matt Toomua and centre Tevita Kuridrani helped sink the Chiefs, while wing Joe Tomane showed great hands to score in the corner.
The Brumbies closed to within one point of competition leaders Coastal Sharks ahead of the Sharks match with the Highlanders in Durban later Friday.
The victory also increased the Brumbies' lead in the Australian conference to six points over the Waratahs, who went down to the Blues in Auckland earlier Saturday.
The Chiefs have now only won twice in 11 trips to Canberra and the Brumbies lead 12-7 overall in their meetings.
The Chiefs attempted to run the ball early, dominating with 70 percent possession to take a 6-3 lead after 25 minutes.
Christian Lealiifano missed some kickable penalties early on before the Brumbies lifted their tempo and held on to the ball.
Flyhalf Toomua burrowed his way over in the 30th minute and Lealiifano landed a penalty for a 13-6 lead at halftime.
Chiefs flyhalf Gareth Anscombe kicked a penalty, but the Brumbies hit back through an acrobatic try in the corner to Tomane to lead 18-9.
The Chiefs launched a series of attacks but the Brumbies defence stood firm and when the New Zealanders looked certain to score Tomane swooped with an intercept and was only prevented from scoring when he was illegally tripped by Lowe.
The numerical disadvantage hurt the Chiefs, with Nic White slotting the ensuing long range penalty goal before Kuridrani's try double gave the Brumbies a 36-9 lead with 18 minutes remaining.
The Chiefs finished with tries to replacements Brad Weber and Josh Hohneck, but an intercept try to Toomua after the siren sunk any hopes of nabbing a bonus point.
"It's an outstanding result for us and we were after a really complete 80 minutes, so for us to do that tonight was really pleasing for the team," Brumbies captain Ben Mowen said.
"Hats off the backline guys because they were outstanding tonight, Matt Toomua at number 10 was really good."
Chiefs skipper Liam Messam said an errors and ill-discipline was costing his team.
"We made some basic errors with the football and our discipline is letting us down along with basic handling errors," Messam said.
"Full credit to the Brumbies. They really stuck it to us and put pressure on us."
The Brumbies cross the Tasman next week for a match against the Crusaders in Christchurch, while the Chiefs play at home against the Lions.
Teams:Brumbies
15 Jesse Mogg, 14 Joe Tomane, 13 Tevita Kuridrani, 12 Christian Lealiifano, 11 Robbie Coleman, 10 Matt Toomua, 9 Nic White, 8 Ben Mowen (captain), 7 Jarrad Butler, 6 Jordan Smiler, 5 Sam Carter, 4 Scott Fardy, 3 Ben Alexander, 2 Stephen Moore, 1 Scott Sio
Substitutes: 16 Siliva Siliva, 17 Ruan Smith, 18 JP Smith, 19 Jack Whetton, 20 Tom McVerry, 21 Michael Dowsett, 22 Andrew Smith, 23 Pat McCabe
Chiefs
15 Tom Marshall, 14 Tim Nanai-Williams, 13 Andrew Horrell, 12 Bundee Aki, 11 Asaeli Tikoirotuma, 10 Gareth Anscombe, 9 Augustine Pulu, 8 Liam Messam (captain), 7 Sam Cane, 6 Nick Crosswell, 5 Brodie Retallick, 4 Ross Filipo, 3 Ben Tameifuna, 2 Mahonri Schwalger, 1 Pauliasi Manu
Substitutes: 16 Nathan Harris, 17 Jamie Mackintosh, 18 Josh Hohneck, 19 Matt Symons, 20 Tevita Koloamatangi, 21 Brad Weber, 22 Dwayne Sweeney, 23 James Lowe
