Membrane-based postcombustion carbon capture can reduce the capture penalty in comparison to absorbent-based separation. To realize this, high-performance membranes are urgently needed with a CO2 permeance exceeding 1000 gas permeation units or GPU, and a CO2/N2 mixture separation factor exceeding 20. Here, we report a new class of organic–inorganic hybrid membranes based on single-layer graphene with a selective layer thinner than 20 nm. For this, the impermeable graphene lattice is exposed to oxygen plasma leading to a high percentage of vacancy defects (porosity up to 18.5%) and is then functionalized with CO2-philic polymeric chains. Treating a gas stream mimicking flue gas, the hybrid membranes yield a six-fold higher CO2 permeance (6180 GPU with a CO2/N2 separation factor of 22.5) than the performance target. Membranes prepared with a combination of optimized graphene porosity, pore size, and functional groups yield a CO2 permeance up to 11 790 GPU. Other membranes yield a CO2/N2 selectivity up to 57.2.