The Brumbies stayed alive in the Super 14 with a 32-12 bonus point victory over the Reds in Canberra extending their winning streak over the Queenslanders to 12 straight wins.
The Brumbies how now won 15 out of the 16 clashes between the teams in Super Rugby.
The Brumbies outscored their opponents by four tries to none.
Alexander is the Brumbies leading try scorer this season and has now scored two tries in one match three times this season.
The Brumbies face the Highlanders in Canberra next week before traveling to Christchurch where they will face the Crusaders.
The Reds face the resurgent Hurricanes in Wellington next week before finishing against the Highlanders in Brisbane.
The Brumbies put in arguably their best performance of the season while the Reds failed to gain any momentum and Cooper and Genia made almost no impact on the game.
The Reds did however lose both of their second rowers in the first quarter of the match and could not win enough ball.
The two-time champion Brumbies kept the Reds try-less in a 32-12 bonus-point victory that has them hovering just two points outside the top four with two rounds left to the playoffs later this month.
It continued the Reds' exasperating run of never having won in Australia's national capital in Super rugby.
The Brumbies, despite the recruitment of high-profile Wallabies Matt Giteau and Rocky Elsom this season, have struggled for application and cohesion but they were in a determined mood from the kickoff against the Reds.
They stamped their authority with two converted tries through forwards Mark Chisholm and Ben Alexander to lead 14-0 in the opening 22 minutes.
They always had the high-flying Reds under pressure, particularly their scrum-base playmakers, Will Genia and Quade Cooper.
"Our attack is starting to take shape and I was very happy with our discipline tonight, particularly in defence," Brumbies' skipper Stephen Hoiles said.
"It has been a frustrating fortnight for us and it's nice to get four tries and play in a nice style."
Prop Alexander scored two barging tries along with others from lock Chisholm and scrum-half Josh Valentine, while Giteau kicked three conversions and two penalties.
The Reds, who were chasing their fourth straight win including impressive efforts against South African heavyweights the Northern Bulls and Western Stormers, were restricted to just four penalty goals from Cooper.
"We lacked spark and they were more enthusiastic than us right from the start," Reds' skipper Genia said.
"We struggled after losing both our locks early and we were severely outplayed."
The Reds were put off their stride with the early injury withdrawals of both locks, Adam Byrnes (knee) and Rob Simmons (cheekbone), while the Brumbies' defence quickly snuffed out their attacking threats.
Cooper gave the Reds hope of coming from behind with four penalty goals to reduce the margin to 14-12 after 51 minutes, but the Brumbies finished the better with tries to Valentine and Alexander to wrap up the contest.
The Reds remained in fourth place, just one point ahead of the Waratahs and two in front of the Brumbies.
The Brumbies must win again with a bonus point against the Highlanders at home next Saturday to maintain their playoff push.
Final Score Brumbies 32 Reds 12
Tries - M Chisholm, B. Alexander 2, J Valentine
Pen - M Giteau 2
Con - M Giteau 3
Pen - Q Cooper 4
Referee: Chris Pollock(NZ)
Assistant referees: Nathan Pearce (Aus), Damien Mitchelmore (Aus)
TMO: George Ayoub (Aus)
Brumbies - 15 Adam Ashley-Cooper, 14 Pat McCabe, 13 Andrew Smith, 12 Tyrone Smith, 11 Brackin Karauria-Henry, 10 Matt Giteau, 9 Josh Valentine, 8 Stephen Hoiles (c), 7 Colby Faingaa, 6 Rocky Elsom, 5 Mark Chisholm, 4 Mitchell Chapman, 3 Salesi Ma'afu, 2 Stephen Moore, 1 Ben Alexander.
Reserves: 16 Huia Edmonds, 17 Jerry Yanuyanutawa, 18 Ben Hand, 19 Michael Hooper, 20 Patrick Phibbs, 21 Robbie Coleman, 22 Ed Stubbs.
Reds - 15 Peter Hynes, 14 Brando Va'aulu, 13 Will Chambers, 12 Anthony Faingaa, 11 Digby Ioane, 10 Quade Cooper, 9 Will Genia (c), 8 Leroy Houston, 7 Daniel Braid, 6 Scott Higginbotham, 5 Adam Byrnes, 4 Rob Simmons, 3 Jack Kennedy, 2 Saia Faingaa, 1 Ben Daley.
Reserves: 16 Sean Hardman, 17 Greg Holmes, 18 Ezra Taylor, 19 Jake Schatz, 20 Ben Lucas, 21 Ben Tapuai, 22 Luke Morahan.