Seeking their first away win of the season, the Brumbies made the most of their rare visits into Stormers' territory with three tries then pulled off a massive 226 tackles to come out on top at DHL Newlands.
Captain Christian Lealiifano described the Brumbies' performance as "an incredible effort from our group".
"[The Stormers] just kept us under pressure the whole time and we couldn't really unlock our backline the way we wanted," Lealiifano said.
"We had to find other ways and our defence was that today, we took our opportunities today when we did get them and the boys definitely saved some as well."
In a controversial end to the first half, a video review after the halftime whistle saw a penalty try awarded to the Stormers with Brumbies winger Toni Pulu earning a yellow card for a high tackle on halfback Herschel Jantjies.
The ruling reduced the Brumbies' advantage to 12-10 with the Stormers taking their first lead early in the second half but Tom Banks put the visitors back in front after 58 minutes and their defence stood tall under immense pressure to close out a famous victory.
The result gives the Brumbies back-to-back wins as they move within reach of the Rebels and Waratahs in the Australian conference ahead of next week's trip to Argentina to play the Jaguares.
Lock Rory Arnold handed the Brumbies a flying start, setting up Pete Samu for the opening try then scoring one of his own from a charge down as they jumped out to an early 12-3 lead.
It appeared that all the Stormers would tally from the first half was a ninth- minute penalty goal to Joshua Stander after passing on multiple chances to kick more points in favour of attacking the Brumbies' line.
But that changed on the stroke of halftime after Pulu's high tackle on Jantjies' 30-metre dash to the line was deemed to deny a legitimate try-scoring opportunity. The Stormers made the most of their one-man advantage to start the second half with Wilco Louw barging over to put the home side in front.
A try-saving tackle from Henry Speight prevented the Stormers from extending their advantage before the Brumbies cashed in again, running through 12 phases for an unmarked Tom Banks to score.
Lealiifano landed the conversion as the Brumbies regained the lead at 19-17 while Stander's penalty miss in the 62nd minute proved costly in the final washup.