The Blues, who had lost each of their first seven matches, looked to be heading towards an eighth straight defeat when they squandered a 13-0 lead to trail 14-13 with less than five minutes to go.
Auckland substitute Ihaia West put the New Zealanders back in front when he calmly converted an angled penalty from near the sideline when the Brumbies were offside at the ruck.
The Brumbies were given a last-gasp chance to steal the points after the siren had sounded when the Blues collapsed a scrum but scrumhalf Nic White missed with his long-range penalty attempt.
"It really was a game of two halves," Blues coach John Kirwan said in a televised interview. "We were outplayed in the second half but we hung on.
"It was a real rollercoaster but we needed that win."
For the Brumbies, it was their third loss of the season but they earned a bonus point to remain in second place on the ladder as the top Australian team.
"I'm really proud of the way we got back into it and gave ourselves a chance to win," Brumbies captain Stephen Moore said.
"We had our chances to win but we just weren't good enough."
The Blues led 13-0 at halftime after flyhalf Daniel Bowden scored all his team's points, two penalties and then a converted try after he sliced through two tired defenders.
Brumbies wing Lausii Taliauli scored a try for the visitors in the second half while Christian Leali'ifano booted three penalties to give the Australians the lead, albeit briefly.
West, who had come on to replace Bowden, was left facing a difficult sideline penalty to restore his team's lead but kept his cool and slotted it between the sticks.
"I was obviously pretty nervous," he said. "It was nerve wracking and I was glad when it went over."
(Writing by Julian Linden in Singapore; editing by Sudipto Ganguly)
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