We propose a new stabilised finite element method for the classical Kol- mogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterised by degener- ate diffusion. The stabilisation is constructed so that the resulting method admits a numerical hypocoercivity property, analogous to the correspond- ing property of the PDE problem. More specifically, the stabilisation is constructed so that spectral gap is possible in the resulting “stronger-than- energy” stabilisation norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behaviour as the “time” variable goes to infinity. We consider both a spatially discrete version of the stabilised finite element method and a fully discrete version, with the time discretisation realised by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical exper- iments verify the theoretical findings