Living systems propagate by undergoing rounds of cell growth and division. Cell division is at heart a physical process that requires mechanical forces, usually exerted by protein assemblies. Here we developed the first physical model for the division of archaeal cells, which despite their structural simplicity share machinery and evolutionary origins with eukaryotes. We show how active geometry changes of elastic ESCRT-III filaments, coupled to filament disassembly, are sufficient to efficiently split the cell. We explore how the non-equilibrium processes that govern the filament behaviour impact the resulting cell division. We show how a quantitative comparison between our simulations and dynamic data for ESCRTIII-mediated division in Sulfolobus acidocaldarius, the closest archaeal relative to eukaryotic cells that can currently be cultured in the lab, and reveal the most likely physical mechanism behind its division.