Tries from Felix Jones, Paul Marshall and Brian Carney inspired Jon Skurr's charges into a 17-0 lead.
With Ireland losing James Coughlan to a yellow card, the Aussies bounced back to take control of the game and move ahead at 21-17.
But Ireland produced a gutsy finish with Brown, who was sprung from the bench, breaking through to score the match-winning try and Tom Gleeson's conversion handed Ireland a three-point win.
Speaking straight after the game, Brown was overjoyed with the victory and his part in it.
"I feel great. We started off the tournament with a big beating from Samoa yesterday so we really needed to start well against Australia.
"It's a great win for us. I only had a bit to do coming on at the end with a few minutes to go but you still get tired in those two minutes."
The result has boosted the Irish camp's confidence ahead of their final Pool D game against Portugal, whom they lost 14-12 to in the semi-finals of last year's Hannover Sevens.
Brown added: "You never know with Sevens. You have just got to keep your composure, especially when you are in the lead.
"You have just got to get control of the ball and then keep it. It can be like a game of tennis, with the ball going back and forth, but we are just delighted that we kept our composure and sealed the win at the end."