In his show Acaster reflects on the best and worst years of his life. His polished storytelling style winning him critical acclaim throughout the festival season.
Other nominees included Anne Edmonds' What's Wrong with You?, Geraldine Hickey's Things are Going Well, Nath Valvo's I'm Happy for You, Tom Allen's Absolutely, and Cassie Workman's Giantess.
The Barry award was named after Barry Humphries, the Australian comedian known worldwide for his drag character Dame Edna Everage.
Humphries came under fire last year for comments he made about transgender people during an interview with Britain's The Spectator."How many different kinds of lavatory can you have? And it's pretty evil when it's preached to children by crazy teachers," he said during the interview.
Melbourne International Comedy Festival director Susan Provan said this week that those comments were not the sole reason the award's name was changed although they "didn't help".
Other awards presented on Saturday included Best Newcomer, won by Blake Freeman for his show There's Something There.
The People's Choice Award, determined by ticket sales, was won by Urzila Carlson, who said this year's festival was the best yet.
Carlson strongly backed changing the name of the award, saying there was currently "very little understanding of transgender and intersex issues."
The Director's Choice Award went to Aaron Chen, the quirky Sydney-sider who many regard as one of rising stars on the Australian comedy scene.
"There were definitely some leaps and bounds for me this festival biggerrooms, more tickets, they let me do the TV gala," he said.
"Next step in my career, I don't know maybe I'll go to Hollywood and become an actor. Hang out with Matt Damon or something."
The comedians' choice Piece of Wood award went to Geraldine Hickey.
The Pinder Prize, which is named after festival co-founder John Pinder and helps support comedians make their Edinburgh Fringe debut, went to Sam Taunton and Steph Tisdell.
Joshua Ladgrove won the "Golden Gibbo" award, which celebrates independent comedians.