<p>Instability analysis is very important for the service safety of functionally graded materials (FGM) sandwich structures. However, there is a great difficulty in identifying the critical points by the existing numerical or analytical methods due to the multiple solutions of nonlinear responses of structures in complex loading conditions. This paper aims to present an analytical method for instability response of FGM sandwich beams in thermo-mechanical loads without a priori assumption of response mode and instability type. Firstly, an analytical method of Emam and Nayfeh is extended to obtain approximate analytic solutions for bending, buckling and snap-through responses of FGM sandwich beams under the framework of strictly satisfying the governing equations and boundary conditions. Secondly, the influence of carbon nanotube (CNT) material component and distribution on the instability condition, type of sandwich beams with functionally graded-CNT reinforced (FG-CNTRC) panels are thoroughly discussed by the bifurcation diagram. In addition, the post-buckling paths of FGM sandwich beams are analytically searched out by the free energy evaluation. The results indicate that the symmetry is broken by the introduction of transverse mechanical load, and this variation makes thermal buckling behavior shift easily from rare bifurcation instability to widespread snap-through instability. By carefully controlling the loading and material parameters, the energy consumption of the deformation jumps can be adjusted, thereby enhancing the resistance to snap-through instability.</p>