The Queensland Reds players, who have won a combined total 105 caps, have not featured for the Wallabies in 2014 because of injury.
Matt Hodgson will captain a side that includes four debutants in head coach Michael Cheika's first match in charge.
Australia face England, Wales, France and Ireland in Tests in November.
Fiji-born winger Henry Speight will make his Australia debut against the Baabaas.
Cooper and Genia haven't played together since last November's 30-26 victory over Wales because of shoulder and ankle injuries respectively.
Cheika was appointed after Ewen McKenzie resigned following defeat by New Zealand in the Rugby Championship this month.
Cheika says this team contains a mix of players from our last game and also experienced players eager to push their way into the Test team.
The Wallabies left Kurtley Beale out of their squad for sending "offensive texts" about a team official.
The Barbarians are coached by Sir John Kirwan and contains 9 New Zealanders in the side to play Australia.
Australia team to face Barbarians: Israel Folau (NSW Waratahs), Henry Speight (ACT Brumbies), Tevita Kuridrani (ACT Brumbies), Matt Toomua (ACT Brumbies), Rob Horne (NSW Waratahs), Quade Cooper (Queensland Reds), Will Genia (Queensland Reds), Ben McCalman (Western Force), Matt Hodgson (Western Force), Scott Higginbotham (Melbourne Rebels), James Horwill (Queensland Reds), Sam Carter (ACT Brumbies), Ben Alexander (ACT Brumbies), Saia Fainga'a (Queensland Reds), Benn Robinson (NSW Waratahs).
Replacements: James Hanson (Queensland Reds), James Slipper (Queensland Reds), Sekope Kepu (NSW Waratahs), Will Skelton (NSW Waratahs), Sean McMahon (Melbourne Rebels), Nic White (ACT Brumbies), Bernard Foley (NSW Waratahs), Christian Leali'ifano (ACT Brumbies), Joe Tomane (ACT Brumbies).
The Barbarians head coach John Kirwan has named nine New Zealanders in his starting fifteen to take on the Wallabies at Twickenham this weekend.
Scotland international Alastair Kellock will skipper the side and will partner Dominic Bird in the second row while there is an all-New Zealand back-row of Adam Thomson, Matt Todd and Steven Luatua.
The back-three consists of Tim Nanai-Williams, Frank Halai and Australian Nick Cummins and they will look to link up with a dynamic centre duo of Francis Saili and South African Juan de Jongh.
Colin Slade starts at first five and the Blues James Parsons starts at hooker.
Barbarians: 15 Tim Nanai-Williams, 14 Frank Halai, 13 Juan de Jongh, 12 Francis Saili, 11 Nick Cummins, 10 Colin Slade, 9 Tomás Cubelli, 8 Steven Luatua, 7 Matt Todd, 6 Adam Thomson, 5 Al Kellock, 4 Dominic Bird, 3 Angus Ta'avao, 2 James Parsons, 1 Matt Stevens.
Replacements: 16 Mahonri Schwalger, 17 Thomas du Toit, 18 Lourens Adriaanse, 19 Heinrich Brussow, 20 Matías Alemanno, 21 Sarel Pretorius, 22 Joaquín Tuculet, 23 Marnitz Boshoff.