In this study, finite element modeling is performed to investigate the compressive failure of the composite sandwich structures with layered composite shells. An embedded debond area between the layered composite shell and the foam core is assumed as a defect. The composite shells are several plies of equal thickness Kevlar, carbon fiber composite, and E-glass composite with epoxy resin. Three different lay-ups, namely, (0°/90°/0°/90°/0°/90°), (45°/−45°/0°/90°/60°/−30°), and (60°/−30°/90°/0°/30°/90°) are considered for symmetric and asymmetric sequences. The work focuses on the importance of cohesive zone model versus the previously conducted numerical simulation and experimental results for buckling of sandwich composite structures. This enables one to account for delamination growth between shells and core and improve the correlation results with those of experiments. It has been shown that not only the cohesive model is capable of demonstrating delamination propagation, but it also correlates very well with the experimental data. By compiling user-defined cohesive mesoscale model in Abaqus simulation, the local and global buckling of the face-sheets can be precisely detected and response of sandwich structure becomes mesh independent, while mesh size is reduced.