Identifying evolution laws of complex systems from massive data has been a challenging issue in the cross field of stochastic dynamics and data science with greater and greater concern in recent years. So far, significant progress has been made in dealing with cases where differential equations are driven by Gaussian noise, yet work for non-Gaussian cases remains great room for growth. In this paper, a data-driven approach based on deep residual networks (ResNets) is proposed to extract governing laws of stochastic dynamical systems perturbed by non-Gaussian α-stable Lévy noise. Specifically, we firstly separate the non-Gaussian noise from drift term for subsequent extraction according to the fundamental theory of differential equations. For the noise part, the Lévy jump measure and noise intensity are estimated via some properties of α-stable distribution. Next the residual network is employed to identify drift. Numerical results of systems of 1 to 3 dimensions are given to demonstrate the feasibility of this method. It is necessary to emphasize here that the contribution of this method is mainly manifested in its effective resolution of the identification for complex non-Gaussian stochastic dynamical systems via cutting-edge machine learning techniques.