A generalized Newtonian potential is derived from the geodesic motion of test particles in Schwarzschild space-time. This potential reproduces several relativistic features with higher accuracy than commonly used pseudo-Newtonian approaches. The new potential reproduces the exact location of the marginally stable, marginally bound and photon circular orbits, as well as the exact radial dependence of the binding energy and the angular momentum of these orbits. Moreover, it reproduces the orbital and epicyclic angular frequencies to better than 6 per cent. In addition, the spatial projections of general trajectories coincide with their relativistic counterparts, while the time evolution of parabolic-like trajectories and the pericentre advance of elliptical-like trajectories are both reproduced exactly. We apply this approach to a standard thin accretion disc and find that the efficiency of energy extraction agrees to within 3 per cent with the exact relativistic value, while the energy flux per unit area as a function of radius is reproduced everywhere to better than 7 per cent. As a further astrophysical application we implement the new approach within a smoothed particle hydrodynamics code and study the tidal disruption of a main-sequence star by a supermassive black hole. The results obtained are in very good agreement with previous relativistic simulations of tidal disruptions in Schwarzschild space-time. The equations of motion derived from this potential can be implemented easily within existing Newtonian hydrodynamics codes with hardly any additional computational effort.