Scalable fabrication of monolayer graphene membrane on porous supports is key to realizing practical applications of atomically thin membranes, but it is technologically challenging. Here, we demonstrate a facile and versatile electrospinning approach to realize nanoporous graphene membranes on different polymeric supports with high porosity for efficient diffusion- and pressure-driven separations. The conductive graphene works as an excellent receptor for deposition of highly porous nanofibers during electrospinning, thereby enabling direct attachment of graphene to the support. A universal “binder” additive is shown to enhance adhesion between the graphene layer and polymeric supports, resulting in high graphene coverage on nanofibers made from different polymers. After defect sealing and oxygen plasma treatment, the resulting nanoporous membranes demonstrate record-high performances in dialysis and organic solvent nanofiltration, with a pure ethanol permeance of 156.8 liters m−2 hour−1 bar−1 and 94.5% rejection to Rose Bengal (1011 g mol−1) that surpasses the permeability-selectivity trade-off.