The Brumbies now have an away game with the table-topping Wellington Hurricanes in New Zealand next Saturday. The other semi-final will see the New South Wales Waratahs host the Otago Highlanders in Sydney.
It is the first time since 2003 that there will be no South African side in the last four.
"It was a special win. I'm so proud of the performance tonight," Brumbies skipper Stephen Moore said at the post-match presentation.
"We knew it was going to be tough but we had a good week and I am so happy to still be alive in the competition.
"It was a good contest, our maul defence was really sharp. We knew that was a strong part of their game and we shut it down reasonably well."
The Australians were the more clinical side in Cape Town, scoring four first-half tries as they led 24-6 at the break.
Tomane showed good speed and power to grab the first three touchdowns, as he was offered too much space out wide, before prop Scott Sio barged over on the stroke of halftime.
Cheslin Kolbe brought the home side back into the contest with an intercepted score but, after making a hash of a number of visits to the Brumbies 22, they were pegged back again when visiting number eight Ita Vaea powered his way over the tryline.
The Brumbies were reduced to 14 men with seven minutes to go when wing Henry Speight received a red card for a dangerous tackle on Juan de Jongh.
They lost another man a minute later when flanker Scott Fardy was sin-binned for collapsing a maul but they still managed to score a sixth try when fullback Jesse Mogg crossed in the corner.
Flyhalf Christian Leali'ifano converted three tries and booted a penalty while opposite number Demetri Catrakilis kicked four penalties and a conversion for the hosts.
"It's very disappointing but credit to the Brumbies, they came out firing. They made it very difficult for us at the breakdown, we couldn't build any phases," said Stormers captain De Jongh.