We investigate diffusion of excitation in one- and two-dimensional lattices with random onsite energies and deterministic long-range couplings (hopping) inversely proportional to the distance. Three regimes of diffusion are observed in strongly disordered systems: ballistic motion at short time, standard diffusion for intermediate times, and a stationary phase (saturation) at long times. We propose an analytical solution valid in the strong-coupling regime which explains the observed dynamics and relates the ballistic velocity, diffusion coefficient, and asymptotic diffusion range to the system size and disorder strength via simple formulas. We show also that in the long-time asymptotic limit of diffusion from a single site the occupations form a heavy-tailed power-law distribution.