However, errors in the closing stages allowed the South Africans to record a bonus-point win in the first away loss for the Brumbies in eight games.
Andries Bekker scored a brace of tries for the Stormers in a week in which he announced he was leaving the franchise for Japan at the end of the Super Rugby season.
"Last week we saw how effective the Brumbies are in the way that they play when they beat the Sharks. To win like that against a quality side was no mean feat," Stormers captain Jean de Villiers said.
After a Joe Pietersen penalty had put them ahead, the Stormers scored the first try of the game through a superb piece of individual skill from Springbok Gio Aplon.
The winger picked up an inside pass from Elton Jantjies and burst through the Brumbies backline. He still had plenty of work to do as he faced Brumbies fullback Robbie Coleman, but through some brilliant stepping he glided past the final defender and crossed the line unopposed.
Christian Lealiifano put the visitors on the board with a penalty but incessant Stormers pressure paid off with another try.
Awarded a kickable penalty, the home side opted to put the ball in the corner instead and were rewarded with Bekker mauling over the line.
The Brumbies hit back with a superb try from Henry Speight. The winger took the ball at pace and broke the Stormers line for a converted score.
The home side extended their lead once more just before halftime when Bekker got his second try. De Villiers picked up the loose ball after a Brumbies' mistake and burst towards the line.
The visitors scrambled back in defence but quick ball at the breakdown put Jantjies five metres out with plenty of options to his right. One of those was the giant Bekker and he had to just take the pass and fall over the line.
After trailing 22-10 at halftime, the Brumbies stepped up in the second, led by the brilliant work of flanker George Smith at the breakdown.
He won both possession and penalties for his side who closed the gap to three points before two Brumbies' mistakes allowed the Stormers to clinch the match.
The home side won a lineout against the throw and forced Fotu Auelua into an error at the breakdown that led to a penalty converted by Pietersen.
The Stormers then scored a bonus-point try as Gerhard van den Heever intercepted Peter Kimlin's wild pass to race in from the halfway line for a converted score.
"We came up against a very desperate Stormers side. We are reasonably happy with this tour, there are a lot of good things we can take from it. We go back to Australia in a reasonable position," Brumbies captain Ben Mowen said.