Post-starburst galaxies are typically considered to be a transition population, en route to the red sequence after a recent quenching event. Despite this, recent observations have shown that these objects typically have large reservoirs of cold molecular gas. In this paper we study the star-forming gas properties of a large sample of post-starburst galaxies selected from the cosmological, hydrodynamical EAGLE simulations. These objects resemble observed high-mass post-starburst galaxies both spectroscopically and in terms of their space density, stellar mass distribution and sizes. We find that the vast majority of simulated post-starburst galaxies have significant gas reservoirs, with star-forming gas masses of ~10$^9$ M$_{odot}$, in good agreement with those seen in observational samples. The simulation reproduces the observed time evolution of the gas fraction of the post-starburst galaxy population, with the average galaxy losing ~90 per cent of its star-forming interstellar medium in only ~600 Myr. A variety of gas consumption/loss processes are responsible for this rapid evolution, including mergers and environmental effects, while active galactic nuclei play only a secondary role. The fast evolution in the gas fraction of post-starburst galaxies is accompanied by a clear decrease in the efficiency of star formation, due to a decrease in the dense gas fraction. We predict that forthcoming ALMA observations of the gas reservoirs of low-redshift post-starburst galaxies will show that the molecular gas is typically compact and has disturbed kinematics, reflecting the disruptive nature of many of the evolutionary pathways that build up the post-starburst galaxy population.
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