Dark energy is frequently modelled as an additional dynamical scalar field component in the Universe, referred to as 'quintessence', which drives the late-time acceleration. Furthermore, the quintessence field may be coupled to dark matter and/or baryons, leading to a fifth force. In this paper, we explore the consequences for non-linear cosmological structure formation arising from a momentum coupling between the quintessence field and dark matter only. The coupling leads to a modified Euler equation, which we implement in an N-body cosmological simulation. We then analyse the effects of the coupling on the non-linear power spectrum and the properties of the dark matter haloes. We find that, for certain quintessence potentials, a positive coupling can lead to significantly reduced structure on small scales and somewhat enhanced structure on large scales, as well as reduced halo density profiles and increased velocity dispersions.