The 1958 Lituya Bay landslide-generated megatsunami is simulated using the Landslide-HySEA model, a recently developed finite-volume Savage-Hutter shallow water coupled numerical model. Two factors are crucial if the main objective of the numerical simulation is to reproduce the maximal run-up with an accurate simulation of the inundated area and a precise recreation of the known trimline of the 1958 mega-tsunami of Lituya Bay: first, the accurate reconstruction of the initial slide and then the choice of a suitable coupled landslide-fluid model able to reproduce how the energy released by the landslide is transmitted to the water and then propagated. Given the numerical model, the choice of parameters appears to be a point of major importance, which leads us to perform a sensitivity analysis. Based on public domain topo-bathymetric data, and on information extracted from the work of Miller (1960), an approximation of Gilbert Inlet topo-bathymetry was set up and used for the numerical simulation of the mega-event. Once optimal model parameters were set, comparisons with observational data were performed in order to validate the numerical results. In the present work, we demonstrate that a shallow water type of model is able to accurately reproduce such an extreme event as the Lituya Bay mega-tsunami. The resulting numerical simulation is one of the first successful attempts (if not the first) at numerically reproducing, in detail, the main features of this event in a realistic 3-D basin geometry, where no smoothing or other stabilizing factors in the bathymetric data are applied.