In a game ruined by the weather, Scotland refused to buckle in the face of a 35-minute siege in the second half before winning a penalty which Greig Laidlaw calmly converted.
"When we had the ball we made the most of it and eventually it came our way," captain Ross Ford said. "It was phenomenal. Australia are a great team and to win away from home is fantastic."
Lashing wind and rain made quality rugby virtually impossible and Scotland, who had the wind behind them in the first half, took charge.
However, their attack looked no more potent against the well-organised Australian defence than it had done in the Six Nations, when they could manage only four tries in five games.
As had been the case all season, Scotland made the breaks with Ross Rennie and Sean Lamont both going through, only for supporting players to spill the passes.
Although Laidlaw missed an early penalty, he did make amends to give Scotland the lead, but by then they had used up more than half their time with the elements behind them.
Laidlaw soon extended the lead with another penalty, but when full-back Stuart Hogg lost control of the ball as he tried to take a tricky low chip, it gave Australia the territory that they needed to collect their first points of the game.
Centre Mike Harris, who was making his debut, put over the kick after the Scotland backs strayed offside.
Scotland suffered a further blow when Lamont picked up a rib injury and was forced off, handing Edinburgh full-back Tom Brown his debut.
With the wind behind them in the second half Australia took instant advantage when Scotland failed to clear the kick-off and Ross Rennie was caught handling in the ruck.
Harris kicked his second penalty to level the scores less than two minutes into the second half.
Australia thought they had taken the lead when they managed to get Rob Simmons, the replacement lock, over the line after rejecting a kick at goal and opting for a lineout a few yards from the Scots line.
The decision went to the television match official but there were no camera angles that showed the ball and Scotland were given the benefit of the doubt.
That set up an extended period of intense Australian pressure as they had the ball consistently within five yards of Scotland's line, but the visitors' defence refused to yield an inch.
It was heroic stuff, but it was also a case of living dangerously with the lineout misfiring badly and preventing Scotland breaking the stranglehold.
Harris and Berrick Barnes both missed with long-range penalties for the home side, while Barnes also missed with an easier drop goal attempt.
Digby Ioane was the dangerman for Australia, but he was once called back for obstruction and the other time he got away he was well shackled by Brown.
Eventually, after making more than 150 tackles, Scotland managed to break the siege and get the ball into the opposition half.
A neat break from Laidlaw set up the attacking position, which went to a series of scrums before the referee lost patience with the Australian pack and handed Laidlaw a simple penalty chance which he gleefully converted.
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