This study explores the impact of nanoscale surface roughness on the stress distribution and concentration along a perturbed solid surface, utilizing the Steigmann–Ogden model of surface elasticity. By applying the corresponding boundary conditions to an isotropic solid with an arbitrary surface shape under plane strain, we aim to account for the coupled effect of membrane and bending surface stiffness. To solve the resulting boundary value problem, we employ the complex variable method and boundary perturbation technique, expressing the unknown functions as power series in a small parameter related to the amplitude-to-wavelength ratio of the undulated surface. This approach enables a reduction of the problem to a recurrent sequence of integral equations with solutions represented as the complex exponential series. A numerical analysis is performed for a cosine-shaped surface using the first-order approximation to examine the influence of undulation wavelength and amplitude, far-field longitudinal stresses, surface tension, axial and bending surface stiffness on the elastic field close to the surface layer.