A refined progressive damage analysis of fiber-reinforced laminated composites subjected to compressive loads is presented here. The numerical analysis exploits higher-order theories developed using the Carrera Unified Formulation, specifically 2D plate theories with Lagrange polynomials to enhance the kinematic approximation through each ply's thickness resulting in a layer-wise structural model. The CODAM2 material model, based on continuum damage mechanics, governs the intralaminar composite damage. The Hashin criteria and the crack-band approach provides failure initiation and propagation, respectively. Fiber micro-buckling and kinking is taken into account via the use of nonlinear post-peak softening models. It is shown that linear-brittle stress-strain softening is effective for accurate compressive strength predictions. A series of numerical assessments on coupon-level composite laminates is carried out to verify the proposed numerical framework while its validation is demonstrated by successfully applying the numerical tool to test cases for which experimental data is available from the literature. Various through-the-thickness structural models are evaluated to provide insights for proper modeling. Numerical assessments considered quasi-isotropic laminates, the compressive strength and size-effects under brittle fracture of notched laminates, and progressive damage characteristics due to stable crack growth in compact compression tests. The results show the possibility of using coarser meshes than those used in standard FEM approaches as the accuracy of predictions is preserved through the use of higher-order structural theories.