Thin-film composite mixed-matrix membranes (TFC-MMMs) have potential applications in practical gas separation processes because of their high permeance (gas flux) and gas selectivity. In this study, we fabricated a high-performance TFC-MMM based on a rubbery comb copolymer, i.e., poly(2-[3-(2H-benzotriazol-2-yl)-4-hydroxyphenyl] ethyl methacrylate)-co-poly(oxyethylene methacrylate) (PBE), and metal–organic framework MOF-808 nanoparticles. The rubbery copolymer penetrates through the pores of MOF-808, thereby tuning the pore size. In addition, the rubbery copolymer forms a defect-free interfacial morphology with polymer-infiltrated MOF-808 nanoparticles. Consequently, TFC-MMMs (thickness = 350 nm) can be successfully prepared even with a high loading of MOF-808. As polymer-infiltrated MOF is incorporated into the polymer matrix, the PBE/MOF-808 membrane exhibits a significantly higher CO2 permeance (1069 GPU) and CO2/N2 selectivity (52.7) than that of the pristine PBE membrane (CO2 permeance = 431 GPU and CO2/N2 selectivity = 36.2). Therefore, the approach considered in this study is suitable for fabricating high-performance thin-film composite membranes via polymer infiltration into MOF pores.