Brown claimed her first Test scalp in her second match for Australia, getting Lauren Winfield-Hill caught in the slips with a ball that swung away.
Aged just 18, Brown she is the youngest player to take a Test wicket for Australia in a men's or women's Test since Holly Ferling did so at at age 17 in 2013.
Ellyse Perry removed Tammy Beaumont lbw for five, with Heather Knight and Nat Sciver the unbeaten batters.
Earlier, Katherine Brunt finished off the third five-wicket haul of her career before Australia's declaration on Friday morning.
Brunt added the scalps of Annabel Sutherland and Jess Jonassen to finish with figures of 5-60.
Her ball to remove Sutherland was arguably the best of the game, moving away from the right-hander and taking her off stump.
The quick also removed Rachael Haynes on 86 on the opening day, helping stunt Australia's momentum after a 169-run fourth-wicket stand between the opener and Meg Lanning (93)Â
Australia then batted through 28 minutes and added just 10 runs in the early session on day two, as questions circulated on whether they should have declared overnight.
Women's Tests are played over four days, with the majority of matches having been drawn given the shorter time frame.
Australia can retain the urn with a win in Canberra, while a draw will leave them needing to win just one of the three ODIs in the multi-format series to keep bragging rights.
History is on Australia's side at Manuka Oval, with no team having lost a women's Test after scoring 300 runs in the first innings.
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