Equilibrium properties of many-body systems with a large number of degrees of freedom are generally expected to be described by statistical mechanics. Such expectations are closely tied to the observation of thermalization, as manifested through equipartition in time-dependent observables, which takes place both in quantum and classical systems but may look very different in comparison. By studying the dynamics of individual lattice site populations in ultracold bosonic gases, we show that the process of relaxation toward equilibrium in a quantum system can be orders of magnitude faster than in its classical counterpart. Classical chaos quantifiers reveal that this is due to a wave packet in a quantum system being able to escape regions of inefficient classical transport by a mechanism akin to tunneling. Since the presented phenomenon takes place in a broad parameter range and persists in weakly disordered systems, we expect that it occurs in a variety of many-body systems and is amenable to direct experimental verification in state-of-the-art quantum simulation platforms.