The result had remained in the balance until Brown's final intervention, although a late drop goal from Berrick Barnes had looked set to see Australia home.
The Wallabies arrived in the East Midlands on the back of successive wins over New Zealand and Wales but they were pushed hard for their latest victory by England's most successful professional club.
Leicester led 15-13 with a quarter of an hour remaining before Barnes and Brown saved Wallaby blushes with a 13-point flurry in the final seven minutes.
Of the 15 players who started Saturday's 25-16 win over Wales in Cardiff, only Matt Giteau and Saia Faingaa were included in the matchday 22, with the Brumbies centre and Reds hooker named among the replacements.
Giteau wasn't called upon despite the close nature of the game, although Faingaa did come on with 15 minutes remaining.
But head coach Robbie Deans did include a number of players who came off the bench at the Millennium Stadium as the Wallabies looked to avoid a similar fate suffered by last year's Springboks, who were beaten 22-17 by the Tigers.
Prop James Slipper, hooker Huia Edwards, scrum-half Luke Burgess, fly-half Barnes and wing Lachie Turner all started for the tourists, with Dean Mumm a second-half substitute.
Australia started the match brightly, racing into a 13-0 lead with just 22 minutes on the clock thanks to a 10th-minute try from Turner and eight points from the boot of Barnes.
A deft offload from Burgess created the space for Turner to power over to set Australia on their way.
Barnes kicked the straightforward conversion to add to an earlier penalty before slotting a second with just over a quarter of the game gone.
Despite their poor start, Leicester hit back through three straight penalties from fly-half Billy Twelvetrees to ensure the gap was just a single point at the break.
The English Champions then took the lead for the first time thanks to another successful penalty from their young outside-half, who missed just one shot at goal in comparison to the less consistent Barnes who was off target on three occasions.
But just as it looked as though Leicester might spring a surprise, the Wallabies showed their class to create three scoring opportunities in the closing stages.
The first was a well-taken penalty from Barnes and the second a drop goal that was made far more difficult due to the poor weather conditions.
Brown's converted try was then the final action of the match as the Wallabies headed for England at Twickenham with an unbeaten tour still a possibility.