The statistical properties of interacting strongly chaotic systems are investigated for varying interaction strength. In order to model tunable entangling interactions between such systems, we introduce a new class of random matrix transition ensembles. The nearest-neighbor-spacing distribution shows a very sensitive transition from Poisson statistics to those of random matrix theory as the interaction increases. The transition is universal and depends on a single scaling parameter only. We derive the analytic relationship between the model parameters and those of a bipartite system, with explicit results for coupled kicked rotors, a dynamical systems paradigm for interacting chaotic systems. With this relationship the spectral fluctuations for both are in perfect agreement. An accurate approximation of the nearest-neighbor-spacing distribution as a function of the transition parameter is derived using perturbation theory.