We report a simple and scalable technique for the fabrication of nanopore arrays on freestanding SiN and graphene membranes based on electron-beam lithography and reactive ion etching. By controlling the dose of the single-shot electron-beam exposure, circular nanopores of any size down to 16 nm in diameter can be fabricated in both materials at high accuracy and precision. We demonstrate the sensing capabilities of these nanopores by translocating dsDNA through pores fabricated using this method, and find signal-to-noise characteristics on par with transmission-electron-microscope-drilled nanopores. This versatile lithography-based approach allows for the high-throughput manufacturing of nanopores and can in principle be used on any substrate, in particular membranes made out of transferable two-dimensional materials.