A Discontinuous Galerkin (DG)-based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium. The proposed method possesses two distinctive features. First, it employs higher-order polynomial-shape functions needed to address the high-frequency propagation regime. Second, it is equipped with curved boundary edges to provide an accurate representation of the fluid–structure interface. The most salient benefits resulting from the latter feature, as demonstrated by the numerical investigation, are the following: (i) an improvement by—at least—two orders of magnitude on the relative error and (ii) the disappearance of spurious resonance frequencies in the surrounding fluid medium. In addition, the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength, the proposed DG method computes the scattered field with a relative error below 1% for an elastic scatterer of about 30 wavelengths. This observation highlights the potential of the proposed solution methodology for efficiently solving mid-frequency to high-frequency elasto-acoustic scattering problems.