The present analysis describes an analytically simple and systematic approximation procedure for modelling the free diffusive spreading of initially super-Gaussian profiles. The approach is based on a self-similar ansatz for the evolution of the diffusion profile, and the parameter functions involved in the modelling are determined by suitable moments of the diffusion equation. The resulting approximate solutions provide an accurate description of the evolution of the initial profile into the Gaussian profile, which is the asymptotically exact solution of the diffusion problem for, in fact, arbitrary initial conditions. (Contains 2 figures.)