We consider a prototypical nonlinear reaction-diffusion system arising in reversible chemistry. Based on recent existence results of global weak and classical solutions derived from entropy-decay related a-priori estimates and duality methods, we prove exponential convergence of these solutions towards equilibrium with explicit rates in all space dimensions.The key step of the proof establishes an entropy entropy-dissipation estimate, which relies only on natural a-priori estimates provided by mass-conservation laws and the decay of an entropy functional.