The buckling analysis of a rectangular sandwich plate smoothly reinforced with multilayered graphene nanoplatelet is presented. The equations of equilibrium are extracted using the principle of minimum total potential energy and the first-order shear deformation theory for the thick sandwich plate. Different graphene distributions are considered through the thickness of the plate, and the rule of mixture is used to obtain the effective properties of the reinforced nanocomposite. To solve the equations, the Fourier series is employed to describe the displacement components and the boundary conditions are considered to be simply supported for all edges. The results are verified with available research, and the effects of some important parameters on the buckling load of the sandwich plate are studied.