Soumya Sarkar brought relief for a anxious Bangladesh by sending back Finch with his fifth ball in what was a soft dismissal off the medium pacer in the 21st over of the innings.
Out of the 20 ODIs that Australia and Bangladesh have played, Australia have won 18, while Bangladesh have been able to win just one.
Warner became only the second batsman to complete 16 ODI tons in 110 innings.
Soumya Sarker broke the stand by surprising Finch (53 off 51) with extra bounce in the 21st over. The openers Aaron Finch and David Warner are in fine form with Steven Smith playing the role of anchor perfectly.
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Bangladesh: Mashrafe Mortaza (capt), Abu Jayed, Liton Das, Mahmudullah, Mehidy Hasan, Mohammad Mithun (wk), Mohammad Saifuddin, Mosaddek Hossain, Mushfiqur Rahim (wk), Mustafizur Rahman, Rubel Hossain, Sabbir Rahman, Shakib Al Hasan, Soumya Sarkar, Tamim Iqbal. Previously, Bangladesh had scored 24 against West Indies in the 38th over. Marcus Stoinis is back from the injury, Zampa is back and so is Nathan Coulter-Nile. Stoinis and Coulter-Nile will play Bangladesh for the first time.
After Warner's dismissal, the 47th over saw the Baggy Greens lose two wickets in quick succession as first Glenn Maxwell (32 off 10) was run out in a mix up with Usman Khawaja at 46.2 overs and then three deliveries later, Khawaja (89 off 72) was caught behind. Australia put away Sri Lanka by 87 runs last Saturday, while Bangladesh swept aside the West Indies by seven wickets on Monday. "There were some half-chances, and especially with the two guys who batted so well for Bangladesh", Sir Viv pointed out.
Mashrafe Mortaza and Mustafizur Rehman largely remained disciplined in their opening spells, bowling wicket-to-wicket.
25 Runs scored by Australia in the 46th over, the moost in a single over in this World Cup.