In this work, we experimentally report the acoustic realization of the two-dimensional Su-Schrieffer-Heeger model in a simple network of air channels. We analytically study the steady-state dynamics of the system using a set of discrete equations for the acoustic pressure, leading to the two-dimensional Su-Schrieffer-Heeger Hamiltonian matrix without use of the tight-binding approximation. By building an acoustic network operating in the audible regime, we experimentally demonstrate the existence of a topological band gap. Moreover, within this band gap we observe the associated edge waves even though the system is open to free space. Our results not only experimentally demonstrate topological edge waves in a zero-Berry-curvature system but also provide a flexible platform for the study of topological properties of sound waves.