Herein, we report a hydrogen-bonded supramolecular hybrid composed of a cobaloxime complex and a covalent organic framework (COF) that achieves photocatalytic overall water splitting. The designed COF features ternary units and dual linkages, which induce asymmetric linkers. Upon photoexcitation, this asymmetry facilitates directed electron transfer and spatially separated redox sites, resulting in a long-lived charge-separated state. Consequently, the COF, hydrogen-bonded with cobaloxime as an H2 production cocatalyst, enables visible-light photocatalytic overall water splitting, simultaneously producing H2 and H2O2, outperforming the COFs with symmetric linker structures. This strategy of asymmetric structural design offers new insights for the design of photocatalysts with suppressed charge recombination for photocatalysis.