Set a revised target of 192 in 41 overs under the Duckworth-Lewis-Stern method, Bangladesh recovered from the early loss of Tanzid Hasan for a duck as Soumya Sarkar and Najmul Hossain Shanto steadied the innings with identical knocks of 42.
After their departures, Mehidy Hasan Miraz anchored the chase with a captain’s knock of 22 not out, holding firm alongside Towhid Hridoy who scored an unbeaten 40, as the pair guided Bangladesh home in 35 overs.
Miraz, who was struck on the helmet by speedster Nathan Ellis towards the end of the chase, went for medical check-ups after scoring the winning runs.
"Winning the series means a lot, but I can't tell you in words, it's an amazing feeling," said Najmul Hossain Shanto, who attended the post-match presentation ceremony in place of Miraz.
"The way we played this series, we showed a lot of courage.
"We have been working really hard the last few months, and we wanted to improve our game. The way we played the last two matches is outstanding, and I'm really proud to be part of this team."
Poor start for Australia
Earlier, Australia, endured their worst-ever start in a 50-overs innings after electing to bat, slumping to three wickets for zero. Marnus Labuschagne led the recovery with a half-century to lift the visitors to 187 for eight in 42 overs before rain interrupted play.
Australia were without captain Mitchell Marsh, who was ruled out of the ODI series due to an ankle injury, while spinner Tanveer Sangha also missed the tour and opener Travis Head was rested.
Stand-in skipper Josh Inglis conceded Australia had been outplayed across the series but said the team would look to sign off on a positive note in the final match.
"We've had guys get starts, but we haven't had anyone go on and make a meaningful contribution," Inglis said at the post-match presentation ceremony.
"We've let ourselves down a bit on that, but Bangladesh have outplayed us so far in this series, so congratulations to them."
The result handed Bangladesh their first ODI series win over the six-time world champions. The third and final match is scheduled for Sunday.
