The separation of azeotropic organic-organic mixtures is critical in the chemical and petrochemical sectors and membrane technology is one of the most successful and sustainable ways to achieve it. A significant challenge is to find materials with the adequate permselectivity for the demanding operational conditions. We propose a carboxyl-functionalized polyimide for pervaporation and separation of polar and non-polar organic azeotropic mixtures with high industrial relevance, such as methanol-toluene, methanol/methyl tert-butyl ether, and ethanol-cyclohexane. Both flat dense films and hollow fibers were investigated. The fabrication as hollow fibers enabled the flux increase by reducing the active layer thickness from 40 to 0.3 μm, while keeping the separation factors nearly unchanged. For methanol/toluene mixtures, by using hollow fibers, the separation factor was around 140, with a flux of 0.54 kg m−2 h−1. For methanol/methyl tert-butyl ether, the separation factor was around 3100 with a flux of 0.14 kg m−2 h−1. For ethanol/cyclohexane, the separation factor was 406 with a flux of 0.24 kg m−2 h−1. The pervaporation performance is a result of a combination of the membrane adequate hydrophilicity and robust mechanical stability, demonstrating their potential application for separating alcohols from organic mixtures.