A late try from Christian Lealiifano which was converted by Stirling Mortlock sealed a 31-32 bonus point victory over the Bulls in Canberra.
Bulls back Olivier scored in the 61st minute and Spies added the side's fourth try five minutes later.
Brumbies captain Stephen Hoiles scored in the corner on the stroke of halftime to give the Brumbies a five-point lead at the break.
With their season on the line, the Brumbies looked to spread the ball early and it paid dividends with a try to blindside flanker Mitchell Chapman in the fourth minute.
Mortlock, playing on the wing in a reworked Brumbies backline, added the extras and another three points four minutes later after the Bulls were caught offside, taking the home side out to an early 10-0 lead.
The Bulls hit back almost immediately when Lealiifano attempted to spread the ball wide, only for it to land in the clutches of winger Bryan Habana, who raced 60 meters to score under the post.
The Brumbies were in again five minutes later after outside center Tyrone Smith, kicking from deep in his own half, found the touch line just five meters out from the Bulls' line.
Lock Peter Kimlin then crossed for a try when the home side managed to win a lineout against the throw.
Mortlock converted to give the Brumbies a 17-7 lead.
But the Bulls came back, replying just over 10 minutes later after exposing the Brumbies' defense out wide and giving fullback Zane Kirchner the space to cross the line for the visitor's second try of the opening half.
"The momentum shifted a fair way during the game, we were very good in patches," said Brumbies captain George Smith.
"We're happy with the win, particularly the four tries and the bonus point."
Bulls captain Fourie du Preez said he was pleased with the way his side came back in the second half.
"It was a great recovery after the way the Brumbies opened against us," said du Preez.
Tries -Mitchell Chapman, Peter Kimlin,Stephen Hoiles,Christian Lealiifano
Penalties -Stirling Mortlock 2
Conversions - Stirling Mortlock 3
Drop Goals -
Cards - Yellow (27) Guy Shepherdson
Tries -Bryan Habana,Zane Kirchner,Wynand Olivier,Pierre Spies
Penalties -Morne Steyn
Conversions -Morne Steyn 4
Drop Goals -
Kick-off: Friday 17 April, 7.40pm (AEST)
Referee: Chris Pollock (New Zealand),
Assistant referees: Paul Marks (Australia), James Scholtens (Australia),
Television match official: Steve Leszczynski (Australia)
Brumbies: Ben Alexander, Stephen Moore, Guy Shepherdson, Ben Hand, Peter Kimlin, Mitchell Chapman, George Smith, Stephen Hoiles, Josh Holmes, Christian Lealiifano, Alfi Mafi, Tyrone Smith, Stirling Mortlock, Adam Ashley-Cooper, Mark Gerrard.
Replacements Huia Edmonds, Salesi Ma'afu, Sitaleki Timani, Julian Salvi, Patrick Phibbs, Gene Fairbanks, Francis Fainifo.
Bulls: Gurthro Steenkamp, Derick Kuun, Werner Kruger, Danie Rossouw, Wilhelm Steenkamp, Dewald Potgieter, Pedrie Wannenburg, Pierre Spies, Fourie du Preez, Morne Steyn, Bryan Habana, Wynand Olivier, JP Nel, Akona Ndungane, Zane Kirchner.