High-index saddle dynamics provide an effective means to compute the any-index saddle points and construct the solution landscape. In this paper, we prove error estimates for Euler discretization of high-index saddle dynamics with respect to the time step size, which remains untreated in the literature. We overcome the main difficulties that lie in the strong nonlinearity of the saddle dynamics and the orthonormalization procedure in the numerical scheme that is uncommon in standard discretization of differential equations. The derived methods are further extended to study the generalized high-index saddle dynamics for nongradient systems and provide theoretical support for the accuracy of numerical implementations.