We calculate light-cone distribution amplitudes for non-relativistic bound states, including radiative corrections from relativistic gluon exchange to first order in the strong coupling constant. We distinguish between bound states of quarks with equal (or similar) mass, m₁ ~ m₂, and between bound states where the quark masses are hierarchical, m₁ >> m₂. For both cases we calculate the distribution amplitudes at the non-relativistic scale and discuss the renormalization-group evolution for the leading-twist and 2-particle distributions. Our results apply to hard exclusive reactions with non-relativistic bound states in the QCD factorization approach like, for instance, B_c → η_cℓν or e⁺e⁻ → J/ψη_c. They also serve as a toy model for light-cone distribution amplitudes of light mesons or heavy B and D mesons, for which certain model-independent properties can be derived. In particular, we calculate the anomalous dimension for the B meson distribution amplitude _B⁻(ω) in the Wandzura-Wilczek approximation and derive the according solution of the evolution equation at leading logarithmic accuracy.