They are two of four changes Arsene Wenger has made to his side after their draw with Southampton on Sunday.
Giroud starts up front in place of Alexandre Lacazette who is named on the bench while Wilshere comes into midfield for the injured Aaron Ramsey.
Ainsley Maitland-Niles makes his first Premier League at left wing-back in place and Alex Iwobi also comes in for Sead Kolasinac.
For the Hammers, David Moyes goes with the same side that beat Chelsea on Saturday.
Adrian continues in goal ahead of Joe Hart while Michail Antonio remains up front with Manuel Lanzini and Marko Arnautovic in support.
West Ham: Adrian, Reid, Cresswell, Zabaleta, Arnautovic, Lanzini, Obiang, Noble, Ogbonna, Masuaku, Antonio.
Subs: Hart, Carroll, Sakho, Hernandez, Collins, Ayew, Rice.
Arsenal: Cech, Koscielny, Sanchez, Wilshere, Ozil, Giroud, Iwobi, Monreal, Bellerin, Xhaka, Maitland-Niles.
Subs: Ospina, Lacazette, Chambers, Welbeck, Kolasinac, Coquelin, Elneny.