In this paper, the development of a space‐time finite element method for the solution of the transient acoustics problem in exterior domains is discussed. The space‐time formulation for the exterior acoustics problem is obtained from a time‐discontinuous Galerkin variational equation for coupled structural acoustics, specialized to the case of zero normal velocities on the wet surface, i.e., a rigid scatterer. The formulation employs a finite computational acoustic domain surrounding the scatterer and incorporates high‐order time‐dependent nonreflecting (radiation) boundary conditions on the fluid truncation boundary as ‘‘natural’’ boundary conditions in the space‐time variational equation, i.e., they are enforced weakly in both space and time. The result is an algorithm for direct transient analysis of acoustic radiation and scattering with the desired combination of good stability and high accuracy. The method is especially useful for the application of adaptive solution strategies for transient acoustics in which unstructured space‐time meshes are used to track wavefronts propagating along space‐time characteristics. Optimal stability estimates and convergence rates are reported together with two representative numerical examples involving transient radiation and scattering which illustrate the high‐order accuracy achieved by the method for the exterior acoustics problem.