In this paper, we reformulate the coupled nonlinear Schrödinger (CNLS) equation by using the scalar auxiliary variable (SAV) approach and solve the resulting system by using Crank-Nicolson finite element method. The fully-discrete method is proved to be mass- and energy-conserved. However, if the convergence results are investigated by using the classical way, the presence of u and v in the equation of r(t) may lead to not only a consistency error of sub-optimal order in time but also some difficulties in analysing the numerical stability. The mentioned difficulties are overcome technically by estimating the difference quotient of the error in the H -norm and carefully analysising the connections of errors between the couple systems. Consequently, the numerical solution is shown to be convergent at the order of O(τ+ h) in the H -norm with time step τ , mesh size h and the degree of finite elements p. Several numerical examples are presented to confirm our theoretical results.