We propose a numerical method for a family of two-dimensional dispersive shallow water systems with topography. The considered models consist in shallow water approximations without the hydrostatic assumption-of the incompressible Euler system with free surface. Hence, the studied models appear as extensions of the classical shallow water system enriched with dispersive terms. The model formulation motivates to use a prediction-correction scheme for its numerical approximation. The prediction part leads to solving a classical shallow water system with topography while the correction part leads to solving an elliptic-type problem. The numerical approximation of the considered dispersive models in the two-dimensional case over unstructured meshes is described, it requires to combine finite volume and finite element techniques. A special emphasis is given to the formulation and the numerical resolution of the correction step (variational formulation, inf-sup condition, boundary conditions,.. .). The numerical procedure is confronted with analytical and experimental test cases. Finally, an application to a real tsunami case is given.