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 5:30 pm local time (1200 GMT) with no prospect of beginning the minimum five-over match before the 6:10 pm 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.
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