Metal-organic frameworks (MOFs), as a peroxidase (POD), can catalyze the conversion of H2O2 to reactive oxygen species (ROS) for antibacterial application. To achieve strong antibacterial activity, it is necessary to improve the enzyme-like activity of MOFs. In this study, FexCoMOF was synthesized by incorporating Co(II) to improve the electron transfer of Fe(III)/Fe(II), which enhanced its redox capacity as a nanozyme and further promoted the generation of hydroxyl radical (⋅OH), exhibiting higher POD activity. Doping with different amounts of Co(II) altered the particle size, specific surface area, and surface defects of FexCoMOF, thus exhibiting differential enzyme-like activities. Additionally, a glucose-responsive enzyme cascade reaction system based on Fe4CoMOF/glucose oxidase (GOx) was established. In the presence of glucose, the in situ-generated substrate H2O2 was in contact with the catalytic site and the generated gluconic acid enabled Fe4CoMOF to maintain maximum enzyme activity under physiological pH conditions, avoiding damage caused by the use of exogenous high concentrations of H2O2. In the in vitro antibacterial experiment, the minimum inhibitory concentration (MIC) of Fe4CoMOF/GOx was 10 μg mL−1 for Escherichia coli (E. coli) and 5 μg mL−1 for Staphylococcus aureus (S. aureus) (∼108 CFU mL−1). The increase in enzyme activity resulted in a considerable reduction in the dose of the antibacterial agent, which was conducive to improving bio-safety. The in vivo experiment in mice demonstrated that the glucose-responsive Fe4CoMOF-based cascade reaction system had excellent antibacterial properties and remarkably promoted wound healing. The antibacterial agent based on bimetallic MOF developed in this work provides a new idea for cascade catalytic antibacterial therapy and will have remarkable application prospects in biomaterials and nanomedicine.