We study the energy-level statistics for electrons in a lateral two-dimensional superlattice in dependence on the strength of the modulation potential at a given perpendicular magnetic field. The interminiband statistics at some selected point in the magnetic Brillouin zone shows a transition from degenerate Landau levels to universal spectral correlations while the corresponding classical dynamics displays a crossover from regular cyclotron orbits to chaotic motion. In terms of the nearest-neighbor spacing distribution, we observe the occurrence of level repulsion for all nonzero modulation amplitudes, which indicates a discontinuous transition to universal fluctuations for the smallest level spacings when the rotational invariance of the system is broken by a finite superlattice potential. In order to test the universality of the transition on larger scales, we propose a random matrix model with one transition parameter for comparison. Confirmed by two different types of potential shapes, we detect an explicit influence of the potential shape on the level spacing distributions, which hence cannot be described by the one-parameter model. Using the spectral rigidity, however, universal long-range fluctuations in the regime of negligible Landau-level mixing are found, in agreement with our random matrix model.