Petaia, 18, will become the youngest Wallabies debutant since James O'Connor a decade ago when he starts on the left wing.
Scrum-half Gordon also gets his chance in Padova, with Cheika making six changes of personnel following the 9-6 defeat to Wales.
Ashley-Cooper will win his first cap for over two years on the right wing, so Israel Folau reverts to full-back.
Matt Toomua comes into the side at 10 with Bernard Foley moving to the centres alongside Samu Kerevi as Kurtley Beale drops to the bench.
Cheika has opted against resting captain Michael Hooper and David Pocock ahead of the showdown with England at Twickenham on November 24.
Full-back Jayden Hayward replaces Luca Sperandio in the only change to the Italy team following their 28-17 victory over Georgia.
Australia: Israel Folau, Adam Ashley-Cooper, Samu Kerevi, Bernard Foley, Jordan Petaia, Matt Toomua, Jake Gordon; Scott Sio, Folau Fainga'a, Taniela Tupou, Izack Rodda, Adam Coleman, Jack Dempsey, Michael Hooper (captain), David Pocock.
Replacements: Tatafu Polota-Nau, Jermaine Ainsley, Sekope Kepu, Rory Arnold, Pete Samu, Will Genia, Kurtley Beale, Dane Haylett-Petty.
Italy: Jayden Hayward, Tommaso Benvenuti, Michele Campagnaro, Tommaso Castello, Mattia Bellini, Tommaso Allan, Tito Tebaldi; Andrea Lovotti, Leonardo Ghiraldini (captain), Simone Ferrari, Alessandro Zanni, Dean Budd, Sebastian Negri, Jake Polledri, Braam Steyn.
Photo Getty Images. Caption: Versatile Australia back Jordan Petaia