Structured catalytic membranes with high porosity and a low pressure drop are particularly suitable for industrial processes carried out at high space velocities. One of these processes is the catalytic total oxidation of volatile organic compounds, which is an economically feasible and environmentally friendly way of emission abatement. Noble metal catalysts are typically preferred due to high activity and stability. In this paper, the preparation of a thermally stable polybenzimidazole electrospun membrane, which can be used as a support for a platinum catalyst applicable in the total oxidation of volatile organic compounds, is reported for the first time. In contrast to commercial pelletized catalysts, high porosity of the membrane allowed for easy accessibility of the platinum active sites to the reactants and the catalytic bed exhibited a low pressure drop. We have shown that the preparation conditions can be tuned in order to obtain catalysts with a desired platinum particle size. In the gas-phase oxidation of ethanol, acetone, and toluene, the catalysts with Pt particle sizes 2.1 nm and 26 nm exhibited a lower catalytic activity than that with a Pt particle size of 12 nm. Catalysts with a Pt particle size of 2.1 nm and 12 nm were prepared by equilibrium adsorption, and the higher catalytic activity of the latter catalyst was ascribed to more reactive adsorbed oxygen species on larger Pt nanoparticles. On the other hand, the catalyst with a Pt particle size of 26 nm was prepared by a solvent evaporation method and contained less active polycrystalline platinum. Last but not least, the catalyst containing only 0.08 wt.% of platinum achieved high conversion (90%) of all the model volatile organic compounds at moderate temperatures (lower than 335 °C), which is important for reducing the costs of the abatement technology.