Having been rescued by the rain last week, London's inclement weather turned the tables on Australia and left them needing to win their last Group A match to avoid going out.
It was the second straight 'no result' for Australia after a similar outcome against New Zealand at Edgbaston on Friday.
They have just two points in the standings, meaning they must beat England in Birmingham on Saturday to have any chance of advancing to the semi-finals.
Bangladesh, who lost to England on Thursday, are also in peril.
They were bowled out for 182 by Australia and then looked set to lose with their opponents handily placed 100 runs from victory with nine wickets in hand and 34 overs remaining when the rain came down.
Australia, needing to bat 20 overs to conjure up a result, sat anxiously through a three-hour wait, willing away the clouds, but a persistent drizzle kept the covers on before the umpires abandoned play.
WARNER QUICK
David Warner had become the quickest Australian batsmen to reach 4,000 ODI runs – achieved in 93 innings -- on his way to an unbeaten 40 with captain Steve Smith not out on 22 when Australia came off after 16 overs at 83-1.
"We were heading down a pretty good path and it was disappointing not to get a result in the end," said Smith.
"The game against England is now a quarter-final as such. They're a good side and the equation is pretty simple now."
Bangladesh won the toss and chose to bat with Tamim Iqbal falling five runs short of a second straight century in the tournament as Mitchell Starc took three wickets in four balls near the end of the innings as they were out in 44.3 overs.
Tamim, who scored 128 against England in the opening game on Thursday, got 95 off 114 balls, picking up the pace and smashing some fierce strokes before being caught by Josh Hazlewood at long leg off Starc with the first ball of the 43rd over.
The Australian paceman bowled Mashrafe Mortaza two balls later, took out the stumps of Rubel Hoosain with his next delivery and came within inches of a hat trick as another swinging yorker flashed past Mustafizur Rahman.
Starc finished with figures of 4-29 in 8.3 overs.
Only two other Bangladesh batsmen -- Shakib Al Hasan (29) and Mehedi Hasan Miraz (14) – reached double figures.
Bangladesh must beat New Zealand in their last group game in Cardiff on Friday and hope England win their next two games in which case they would accompany the hosts to the semi-finals.
"Australia were unlucky and we are lucky to still be in the tournament. We have to play hard again, but we beat New Zealand in Ireland (in a warm-up match) so you never know," added Bangladesh skipper Mortaza.
(Writing by Mark Gleeson; Editing by Ken Ferris)
