Wales had been 34-13 ahead after 55 minutes but then saw Australia fight their way back for a memorable win - only their fifth in 14 internationals this year.
Winger Mark Nawaqanitawase (22) scored two tries to lead the comeback as the Wallabies outscored Wales five tries to four.
Defeat for the hosts followed the embarrassment of last week’s home loss to Georgia, which increased speculation about Pivac's future less than a year from the World Cup in France.
Two tries in the space of 10 minutes from Nawaqanitawase, who had won his first cap just a fortnight ago, sparked the comeback after Wales had been 20-13 ahead at the break and then stretched their lead with two early second-half tries.
But they were overwhelmed in the closing stages as Australia followed Nawaqanitawase’s double with a penalty try six minutes from time and went ahead on the stroke of full-time, with Nawaqanitanwase again at the heart of the action before replacement hooker Lachlan Lonergan (23) walked in the winning try.
That gave Australia a second win on their five-test tour of Europe and offered relief to coach Dave Rennie, but for Pivac the pressure is sure to mount as Wales lost for a third time in their four autumn internationals.
They were comprehensively beaten by New Zealand at the start of the month, defeated Argentina but then lost to Georgia for the first time.
Their 34 points were the most the Welsh have scored against Australia but that was not enough despite a bright start.
Veteran lock Alun Wyn Jones (37) showed a handy swerve to set up Jac Morgan (22) for Wales’ first score in the 10th minute, followed 12 minutes later by a try for Taulupe Faletau (32) on his 100th test appearance.
But Australia's Folau Fainga'a (27) crashed over from a maul to reduce the deficit to seven points at halftime.
Wales raced into a 21-point lead within 12 minutes of the restart as Morgan and Dyer both added tries.
Nawaqanitawase’s 58th-minute try suddenly offered Australia some hope and the winger, who made his debut on the tour, scored again 10 minutes later from an attacking line-out as Lonergan played an inside pass to the wing to score.
Then the penalty try, awarded when Ryan Elias (27) was adjudged to have deliberately collapsed the driving maul, brought Australia within two points and, with two Wales players in the sin bin, the Wallabies scored again for a memorable test win.
