The characteristics of grain boundaries (GBs) and their mechanical responses under external loading are very important in governing the strength and plasticity of polycrystalline ceramics. In this study, first-principles calculations are used to investigate the stability of Σ5310001 GBs in (HfNbTaTiZr)C high-entropy carbide ceramic (HECCs) and its constituent binary transition-metal carbides (TMCs), as well as their mechanical behaviors under shear and tensile deformations. The results show that the Σ5310001 GBs in all systems are classified as “Open GB” and “Compact GB” according to their morphologies, with the Open GB exhibiting lower GB formation energy, thereby possessing greater structural stability. Under shear deformation, all carbides display shear-coupled GB migration, except for the Open GBs in group IVB TMCs, where the formation of C—C bonds induces supercell failure through the rupture of TM—C bonds. Furthermore, the initial migration stress of Open GB in the HECC is higher than that in binary TMCs, highlighting the strengthening effect introduced by multicomponent GBs. Under tensile deformation, binary TMCs containing Compact GB primarily fail through graphitization, whereas the HECC exhibits both graphitization and intergranular fracture. For the Open GB, group IVB TMCs yield due to increased excess volume of GB, while group VB TMCs undergo intergranular fracture; both failure mechanisms coexist in the HECC. Notably, the HECC containing Compact GBs exhibits yield strength comparable to the peak strength of binary TMCs, exceeding the “weakest-link” limit typically associated with ideal condition (0 K and defect-free). Overall, this study elucidates the synergistic roles of GB and multicomponent effects in governing mechanical responses in HECC, indicating that the interplay between multicomponent effects and defects may be the basis for the exceptional mechanical performance of high-entropy materials. These findings provide theoretical guidance for GB engineering and mechanical optimization in HECCs, and they offer insights into exploring their mechanical behaviors under complex defect interactions.