Gallagher, who was ineligible to face his parent club at Stamford Bridge, starts in a three-man midfield alongside Cheikhou Kouyaté and James McArthur.
Andersen, who joined from Lyon this summer, partners fellow new signing Marc Guéhi in the centre of defence, with Tyrick Mitchell and Joel Ward on either side.
In attack, Christian Benteke makes his first start of the season after coming on as a second-half substitute against Chelsea. Wilfried Zaha and Jeffrey Schlupp are alongside him, with last weekend's debutant Jesurun Rak-Sakyi among the substitutes.
Brentford are unchanged from the side that beat Arsenal last weekend, with goalscorers Sergi Canós and Christian Nørgaard in from the start. Ivan Toney leads the line once again.
Palace: Guaita (GK), Ward, Guéhi, Andersen, Mitchell, Kouyaté, McArthur, Gallagher, Zaha, Schlupp, Benteke.
Subs: Butland (GK), Matthews (GK), Tomkins, Ayew, Mateta, Kelly Riedewald, Rak-Sakyi.
Brentford: Raya (GK), Henry, Pinnock, Nørgaard, Canós, Onyeka, Toney, Jansson, Mbeumo, Ajer, Janelt.
Subs: Fernández (GK), Goode, Forss, Wissa, Ghoddos, Dervisoglu, Bidstrup, Sørensen, Roerslev.