Agar was part of Australia's squad for the Test tour of India but didn't play a game, while Cartwright missed the series but got his baggy green in the Third Test against Pakistan at the SCG in January.
National Selector Trevor Hohns said: "Ashton has continued to impress us with his form and we believe his bowling is at a level where he deserves to be playing on the highest stage.
"He will work nicely in tandem with Nathan Lyon and also brings a great all-round package to the team.
"Whilst Steve O'Keefe bowled well in Pune, he did not maintain this level in the remaining matches of the series and we believe the timing is right for Ashton to enter the set-up and test his all-rounder ability.
"Hilton averages nearly 60 runs in first class cricket and was the second highest run scorer in the Sheffield Shield last season with 861 runs.
"He is a quality player who we believe has a big future for Australia and we are very keen to see him carry on his good form into sub-continent."
Shaun Marsh, who played all four Tests in India, isn't part of the squad, along with the injured Mitchell Marsh.
Australia will arrive in Bangladesh on August 18 with the two Tests to be played in Dhaka from August 27 and Chittagong from September 4.