This work presents a numerical method for the simulation of landslides generated impulse waves and its application to the historical Vajont case study. The computational tool is based on the Particle Finite Element Method (PFEM), a Lagrangian strategy that combines the finite element solution of the governing equations with an efficient remeshing strategy to deal with large deformation problems. After presenting the numerical formulation, different landslide impulse wave problems with Froude number ranging from 0.5 to 2.8, are analyzed to validate the proposed methodology. The computational method is shown to be able to reproduce accurately the landslide runout, the momentum transfer between the sliding material and the impounded water, and the consequent wave propagation observed in experimental physical models. Then, the PFEM model is applied to the numerical simulation of the Vajont disaster, which is analyzed with a fully-resolved three-dimensional model. The numerical results are discussed and compared to the post-event observations and the numerical results of other computational methods. The results in terms of landslide velocity and runout, geometry of the deposit, maximum water runup, dam overtopping wave, and water discharge in the downstream valley are in good agreement with observations and reconstructions. The calibration and validation performed for this study form the basis for the PFEM analyses presented in a companion paper finalized to simulate different scenarios of the Vajont rockslide considered in the experimental tests done a year before the disaster.