After being bowled first ball in the series opener, Matthew Short survived just three deliveries before Taskin Ahmed knocked over his stumps.
Mustafizur Rahman then struck twice in the following over, removing Cooper Connolly and Matt Renshaw, both caught behind by Litton Das.
Australia's top-order collapse marked the first occasion they had lost three wickets without a run on the board in an ODI.
Only three other teams had previously suffered the same fate in ODI cricket. Bangladesh lost three wickets in the first three balls against Sri Lanka during the 2003 World Cup, while Pakistan endured similar collapses against South Africa in 1997 and New Zealand in 1983.
The early breakthroughs handed Bangladesh a dream start as they looked to build on their victory in the series opener and clinch the three-match ODI series.
