Bresciano's thumping 30-yard strike put the Socceroos in front, but it was a lead they held for only 10 minutes - with the impressive Jordan Rhodes levelling with a glancing header.
Scotland capitalised on their superiority in the second half and went ahead courtesy of a headed own goal by unfortunate debutant Jason Davidson, barely two minutes after coming off the bench.
Substitute Ross McCormack sealed the win with a low drilled effort 14 minutes from time.
Scotland came flying out of the blocks and Rhodes, making his first senior start for Scotland, could have put the home side ahead in the first few minutes when he collected Robert Snodgrass’s cross from the right – but was muscled out of it by the Australia defence
The home side continued to threaten, but Australia bagged the opener in controversial – yet emphatic - style.
A series of refereeing errors saw the Socceroos denied a penalty when Robbie Kruse hooked the ball into the six-yard area despite it being a yard over the byline.
Play was waved on and Australia squeezed in a shot at goal – only for Danny Fox’s clear handball on the line to go unnoticed.
Australia were perplexed at the lack of a penalty,but they responded in the best way possible. Seizing on Scotland’s half clearance of the resulting corner, Bresciano thrashed the dropping ball into the net with a stunning volley – a goal the Al Nasr midfielder may never top. It was his first international goal in four years – quite a way to end the drought.
But Scotland responded well against a tentative Australia side and deserved their equaliser when it arrived courtesy of Rhodes, who used the pace of Fox’s cross to nod the ball into the far corner.
Scotland pressed for a second and it eventually came via the head of the luckless Davidson, who appeared to be caught in two minds before heading beyond his own goalkeeper Adam Federici.
The game became disjointed after both sides made wholesale second half changes, but one of those to impress from the bench was Ross McCormack.
The Leeds striker could have settled the match when he raced clean through – only to be wrongly flagged offside.
But he wasn’t to be denied moments later, as he drove into a shooting position on the edge of the Australia box before firing low into the bottom corner.
The win may prove a confidence-booster ahead of Scotland's upcoming World Cup qualifiers against Serbia and Macedonia.