In this paper, we propose a coupled discontinuous Galerkin (DG) and continuous Galerkin (CG) scheme for solving the nonlinear evolution Davey-Stewartson (DS) system in dimensionless form. The DS system consists of two coupled nonlinear and complex structure partial differential equations. The wave’s amplitude in the first equation is solved by the high-efficiency local DG method, and the velocity in the second equation is obtained by a standard CG method. No matching conditions are needed for the two finite element spaces since the normal component of the velocity is continuous across element boundaries. The main strengths of our approach are that we combine the advantage of DG and CG methods, using DG methods handling the nonlinear Schrödinger equation to obtain high parallelizability and high-order formal accuracy, using the continuous finite elements solving the velocity to maintain total energy conservation. We prove the energy-conserving properties of our scheme and error estimates in $L^2$-norm. However, the non-linearity terms bring a lot of trouble to the proof of error estimates. With the help of energy-conserving properties, we construct a series of energy equations to obtain error estimates. Numerical tests for different types of systems are presented to clarify the effectiveness of numerical methods.