In this paper, the influence of an elastically graded material on selected non-slipping adhesive contacts under combined normal and tangential loading and plane strain conditions is investigated. These include the problem of an adhesive contact between a power-law graded substrate and a rigid parabolic cylinder loaded by an inclined pull-off force, for which closed-form analytical solutions are derived. While the maximum pull-off force in terms of magnitude occurs either under purely normal or tangential loading (in the latter case, peel-off instead of pull-off is present), for materials characterized by a higher positive in-depth grading its minimum is reached under inclined pull-off. The corresponding pull-off angle depends on both the material grading and the Poisson’s ratio. A further application focuses on the effect of material grading on adhesive contacts subjected to an increasing tangential force at a fixed normal force. The shear-off force, at which the transition from complete stick to a different friction mechanism arises, as well as the associated shear-induced area reduction is analyzed in detail. The results are valuable for predicting adhesion and detachment in mechanical and biomechanical systems with functionally graded materials.