Leg-spinner Tanveer Sangha (hamstring) will also be absent from the three-match series starting in Dhaka on Tuesday.
Marsh will, however, fly over to be with the team as he prepares for a comeback in the subsequent Twenty20 series in Bangladesh.
Teenage batsman Ollie Peake and big-hitter Matt Short, who were due to head home after Pakistan, will now stay on.
Off-spinner Todd Murphy, who has seven Test caps but is yet to play limited-overs cricket for Australia, will fill Sangha’s role.
“Unfortunately, Tanveer Sangha sustained a hamstring injury during the second ODI against Pakistan and will take no further part in the tour,” said national selector Tony Dodemaide.
“We were hopeful Mitch Marsh would be available for the Bangladesh ODI series however he is still returning to full fitness from an ankle injury.
“Travis was initially selected in the ODI and T20I legs of this Bangladesh tour but has subsequently been granted personal leave for both,” he added.
Australia squad: Josh Inglis (C), Xavier Bartlett, Alex Carey, Cooper Connolly, Ben Dwarshuis, Nathan Ellis, Cameron Green, Matthew Kuhnemann, Marnus Labuschagne, Todd Murphy, Oliver Peake, Matthew Renshaw, Liam Scott, Matt Short, Adam Zampa
