In this paper we consider a general class of stochastic differential equations driven by $\alpha$-stable processes with Lipschitz drift coefficients and non-Lipschitz diffusion coefficients. An Euler-Maruyama approximate solution is proved whenever the diffusion coefficient is H\"{o}lder continuous with exponent $\beta < \frac{\alpha}{2}$ with $\alpha \in (1,2)$. We obtain a strong rate of convergence of order $ n^{-\beta^{2}}$. 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 L\'evy-It\^{o} 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.