The home side, celebrating Irish rugby's birthday, were 13-5 down at the break but rallied back from an error-filled first half to post a narrow success.
Ireland’s points came from tries by Josh van der Flier, captain Caelan Doris and substitute Gus McCarthy, with new flyhalf Sam Prendergast contributing a penalty and conversion and his replacement Jack Crowley a conversion.
Max Jorgensen scored Australia’s lone try, with Noah Lolesio kicking over four penalties and a conversion, as they lost for a second straight week after Scotland beat them last Sunday.