Gas separation membranes based on single-layer-graphene are highly attractive because the size of graphene nanopores can be tuned to separate gases by the size-sieving mechanism. A prerequisite for this, the synthesis of high-quality polycrystalline single-layer graphene film by chemical vapor deposition (CVD), is extremely crucial. The quality of graphene in the context of membranes is reflected by the size and the density of the intrinsic vacancy defects, and is affected by the catalytic metal substrate and the CVD environment. Generally, expensive high-purity Cu foil is used to obtain gas-sieving performance from single-layer graphene. For the eventual scale-up of graphene membranes, it is highly attractive to use low-cost Cu foils, however, as we show here, these Cu foils are rough and graphene membranes derived from these foils do not yield gas-sieving performance. Herein, we conduct a systematic high-temperature annealing study on two separate, commercial, low-cost Cu foils leading to their transformation to Cu(111). The annealing process smoothened the Cu surface, decreasing the root mean square (RMS) surface roughness from over 200 nm to close to 100 nm. The RMS roughness on the individual Cu step, measured using the scanning tunneling microscopy (STM), was only 0.23 nm. The smooth, oriented Cu grains yielded single-layer graphene with a significantly lower defect density with ID/IG ratio decreasing from 0.18 ± 0.02 to 0.04 ± 0.01. Finally, single-layer graphene films, synthesized on the annealed low-purity Cu foil, yielded H2-selective membranes with H2 permeance reaching 1000 gas permeation units (GPU) in combination with attractive H2/CH4 and H2/C3H8 selectivities of 13 and 26, respectively.