Two tries in a four minute second-half spell at the Free State Stadium turned around a 11-9 halftime deficit and handed the Cheetahs their third win in 13 matches this season.
A break from the side of the scrum set up a try for wing Willie Le Roux eight minutes into the second half followed quickly by a leaping catch from Goosen to go over for the second.
Goosen, who switched during the game from flyhalf to fullback, kicked four penalties and a conversion, as well as a drop-goal from inside his own half near fulltime.
The Brumbies scored the only try of the first half through Robbie Coleman and got a second late in the game through Stephen Moore.
(Reporting by Mark Gleeson; editing by Josh Reich)
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