In this paper we consider a general class of stochastic differential equations driven by stable processes with Lipschitz drift coefficients and non-Lipschitz diffusion coefficients. An Euler-Maruyama approximate solution is proved whenever the diffusion coefficient is Holder continuous with given exponent. We obtain a strong rate of convergence. Our proposed method is new in this context and is based on a truncation method by separating the large and small jumps of the $\alpha$ stable process in the Lévy-Itô decomposition. Along the paper we give some numerical simulations of stochastic models that match our results, namely some stable driven Ornstein-Uhlenbeck, Cox-Ingersoll-Ross and Lotka-Volterra type processes.