  Twisted moir\'e superlattices (TMSs) are fascinating materials with exotic
physical properties. Despite tremendous studies on electronic, photonic and
phononic TMSs, it has never been witnessed that TMSs can exhibit higher-order
band topology. Here, we report on the experimental observation of higher-order
topological states in acoustic TMSs. By introducing moir\'e twisting in bilayer
honeycomb lattices of coupled acoustic resonators, we find a regime with
designed interlayer couplings where a sizable band gap with higher-order
topology emerges. This higher-order topological phase host unique topological
edge and corner states, which can be understood via the Wannier centers of the
acoustic Bloch bands below the band gap. We confirm experimentally the
higher-order band topology by characterizing the edge and corner states using
acoustic pump-probe measurements. With complementary theory and experiments,
our study opens a pathway toward band topology in TMSs.
