We carry out the convergence analysis of the Scalar Auxiliary Variable (SAV) method appliedto the nonlinear Schrödinger equation which preserves a modified Hamiltonian on the discretelevel. We derive a weak and strong convergence result, establish second-order global error boundsand present long time error estimates on the modified Hamiltonian. In addition, we illustrate thefavorable energy conservation of the SAV method compared to classical splitting schemes in certainapplications.