Electrochemical supercapacitors (SCs) are high-efficiency electrochemical energy storage devices that can deliver energy at a very fast rate. Metal–organic framework (MOF)-derived layered double hydroxides (LDHs) are promising materials with great potential for commercial SC applications. In this study, a two-step synthetic strategy was developed to produce porous nanostructured ZnCoNi-LDH nanosheets (NSs) from bimetallic MOFs through a chemical reduction method. All the as-synthesized materials were characterized for their crystal structure, phase, morphology, and surface construction. The optimized ZnCo2Ni-LDH composition exhibited the best charge storage ability and specific capacitance among the different as-synthesized compositions. The specific capacitance of the fabricated device ZnCo2Ni-LDH||cellulose paper-KOH||ZnCo2Ni-LDH was found to be 348.2 F g–1 at 1.0 A g–1, with a maximum energy density of 54.4 W h kg–1 and a power density of 4439.0 W kg–1. After 10,000 continuous charge–discharge cycles, the device retained 86% of its capacitance. It also delivered a high-capacitance-specific Coulombic efficiency (57–60%). A mechanistic study corroborated the experimental results that the high pore volume, possible insertion of a greater number of electrolyte ions, multioxidation states of Ni and Co ions, and their synergistic effect with Zn2+ all contributed to the diffusion-controlled charge storage behavior. The charge storage characteristics of the device were found to be a combination of redox and electrostatic effects, forming a facile hybrid SC.