Zimbabwe joined Sri Lanka as qualifiers from Group B and will move on to face the West Indies, India and South Africa in the next phase.
Ireland were also eliminated.
Persistent drizzle all day at the Pallekelle Stadium kept the covers on and the umpires finally called it off at 5.30pm local time with no prospect of beginning the minimum five-over match before the 6.10pm cut-off time.
Australia, who had been beaten by Sri Lanka and Zimbabwe, needed an Ireland victory to keep their hopes alive.
Zimbabwe are now on an unassailable five points with a game to still to come, while Ireland are third on three points.
Australia, on two points, have one game remaining against Oman on Friday, but cannot overtake Zimbabwe.
Australia had a shambolic build-up to the tournament.
They lost leading pace bowlers Pat Cummins and Josh Hazlewood to injury, while captain Mitchell Marsh missed the first two games after being hit in the groin in training and suffering testicular bleeding.
Test great Steve Smith was left out despite being in terrific form top of the order in the Big Bash League and his expertise in playing against spin bowling.
He flew in as cover last week and was formally added to the Australia squad at the weekend.
But then he was left out for the must-win game against Sri Lanka on Monday which saw an Australia middle-order collapse as they lost by eight wickets.
Zimbabwe captain Skiander Raza hailed his side’s achievement in reaching the last eight, but warned there were tougher battles to come.
“We have come this far. We will celebrate but we have a game against Sri Lanka to come,” said Raza.
“This is only a tick in the box and this is not the only thing that we set out for. Everyone loves an underdog story.”
Ireland skipper Lorcan Tucker was downcast after his side did not get to play.
“Obviously very disappointed to go out in this fashion, but completely uncontrollable,” said Tucker.
“Our batting failed. There were big moments in the tournament particularly against Sri Lanka when we dropped so many catches.
“Hopefully, we will handle these tough situations differently moving forward.” (AFP)
Edited by Tony Sabine
