The contact behavior of an elastic film subjected to a mismatch strain on a finite-thickness graded substrate is investigated, in which the contact interface is assumed to be nonslipping and the shear modulus of the substrate varies exponentially in the thickness direction. The Fourier transform method is adopted in order to reduce the governing partial differential equations to integral ones. With the help of numerical calculation, the interfacial shear stress, the internal normal stress in the film and the stress intensity factors are predicted for cases with different material parameters and geometric ones, including the modulus ratio of the film to the substrate, the inhomogeneous feature of the graded substrate, as well as the profile of the contacting film. All the physical predictions can be used to estimate the potential failure modes of the film–substrate interface. Furthermore, it is found that the result of a finite-thickness model is significantly different from the prediction of a generally adopted half-plane one. The study should be helpful for the design of film–substrate systems in real applications.