Acoustic black holes (ABHs) have recently been revealed as an effective vibration-control technology. However, owing to their disadvantages such as low stiffness, structural discontinuity and manufacturing difficulties, ABHs are limited in terms of their practical application. Therefore, a multilayer composite sandwich plate with filled imperfect-ABH (F-IABH) is proposed in this study to attenuate the vibration and sound radiation of the multilayer sandwich plate, as well as to enhance its bearing capacity and maintain its topological continuity. The structural-damping characteristics and vibroacoustic response of F-IABH are determined using a semi-analytical wave propagation model developed based on symplectic and discrete radiation model methods. The accuracy and effectiveness of the analytical model are validated using the finite-element method. The results show that the F-IABH can achieve broadband vibration-attenuation and noise-reduction effects, and their vibration and noise reduction effects are significantly better than those of ABH structures. Additionally, the wave-damping characteristics of uniform, ABH, and F-IABH plates are analyzed. The results show that the wave damping of the F-IABH plate is much higher than that of the uniform plate, and that it generates more propagative waves to dissipate energy. Additionally, the damping-attenuation characteristics of each wave in the F-IABH and ABH plates exhibit similar variations. However, owing to the large variation range of the wave-mode damping characteristics of the F-IABH plate, the control effect of the latter on structural vibration and acoustic radiation is more significant than that of the ABH plate.