In the development of adhesives, an understanding of the fracture behavior of the bonded joints is inevitable. Two typical failure modes are known: adhesive failure and cohesive failure. However, a molecular understanding of the cohesive failure process is not as advanced as that of the adhesive failure process. In this study, research was developed to establish a molecular understanding of cohesive failure using the example of a system in which epoxy resin is bonded to a hydroxyl-terminated self-assembled monolayer (SAM) surface. Adhesive failure was modeled as a process in which an epoxy molecule is pulled away from the SAM surface. Cohesive failure, on the other hand, was modeled as the process of an epoxy molecule separating from another epoxy molecule on the SAM surface or breaking of a covalent bond within the epoxy resin. The results of the simulations based on the models described above showed that the results of the calculations using the model of cohesive failure based on the breakdown of intermolecular interactions agreed well with the experimental results in the literature. Therefore, it was suggested that the cohesive failure of epoxy resin adhesives is most likely due to the breakdown of intermolecular interactions between adhesive molecules. We further analyzed the interactions at the adhesive failure and cohesive failure interfaces and found that the interactions at the cohesive failure interface are mainly accounted for by dispersion forces, whereas the interactions at the adhesive failure interface involve not only dispersion forces but also various chemical interactions, including hydrogen bonds. The selectivity between adhesive failure and cohesive failure was explained by the fact that varying the functional group density affected the chemical interactions but not the dispersion forces.