Davies, a left-arm spinner, returned figures of 3-17 in three overs while fellow slow-bowler took 3-18 as the visitors limited the tigers to a total of 131 all out after the hosts elected to bat.
The Bangladesh batting effort failed to fire as top score came from number 8 Mahedi Hasan with 29 not out after the home team slipped to 78-6 inside 12 overs.
Australia's spinners dominated in the first innings with Matt Renshaw also get into the action with 2-26 while another debutant Nikhil Chaudhary took 1-14 in two overs.
Cooper Connolly got the the run-chase off to a rapid start with 47 from 27 balls but the tourists kept losing wickets as continued to take on the home side's slow-bowlers.
Davies completed a good a day out when he was unbeaten on 7 when the winning runs were scored in the 19th overs as Australia took a series lead heading into the remaining games.
Bangladesh ODI captain Mehidy Hasan Miraz says healthy competition in the squad and a conscious effort to prepare better pitches at home has contributed to their improved form.
Australia veteran spinner Nathan Lyon is confident he will be ready for the Test series at home against Bangladesh in August as he continues his recovery from what he describes as a "traumatic" injury.
Cooper Connolly produced the finest innings of his international career to drag Australia to a thrilling one-wicket victory over Bangladesh in the third ODI in Mirpur, preventing a series whitewash despite a remarkable late collapse.
South Africa's Women's T20 World Cup campaign began with a disappointing 65-run defeat to Australia as a record chase proved beyond them.
