As playmaker Bernard Foley admitted, the Wallabies were their own worst enemies during a dreadful first half.
They trailed 17-12 at the break, gifting the Scots all of their points through an early penalty goal, then an intercept try to centre Duncan Taylor and a charge-down effort from goalkicking five-eighth Finn Russell.
Only Folau's two strikes kept the Wallabies in the contest.
Foley set both of them up for his NSW Waratahs teammate either side of a stint in the sin bin for a late hit on Russell when Scotland's inspirational No.10 broke free with some innovative counter-attack from his own quarter.
First, Foley found Folau with a long ball in one of the Wallabies' rare forays deep in Scotland territory.
Then, on the stroke of halftime, Folau soared high above Scotland fullback Gordon Reid to reel in Foley's pinpoint crossfield kick to give the Wallabies momentum heading into the second section.
But despite being a man down with No.8 Ryan Wilson yellow-carded a minute before halftime, the Scots refused to yield until Australia briefly snatched the lead in the 57th minute.
Foley's successful conversion of a desperate dive-over try from Will Genia earned the Wallabies a 19-17 advantage.
But, a 50-metre try three minutes later to flanker Hamish Watson restored the visitors' five-point lead and, despite several opportunities, the Wallabies were unable to find a reply to give Foley another chance to break Scottish hearts.
Last-gasp penalty goals from Foley had denied the Scots at the 2015 Rugby World Cup at Twickenham and again on the Wallabies' spring tour last November.
SCOTLAND 24 (Finn Russell, Duncan Taylor, Hamish Watson tries Russell 3 cons Greig Tonks pen) bt AUSTRALIA 19 (Israel Folau 2, Will Genia tries Bernard Foley 2 cons) at Allianz Stadium. Referee: Wayne Barnes. Crowd: 30,720.