The paper analyzes a system decoupling approach over the entire time domain to solve a diffusive Peterlin viscoelastic model which is a type of non-Newtonian fluids. The main idea is decoupling this non-Newtonian fluid equations into Newtonian fluid equations and second-order quasilinear parabolic equations entirely in the time domain, which can be solved iteratively. Under this approach, existing excellent computational method interfaces for Newtonian fluid equations and parabolic equations can be directly utilized. By employing the weak form of the Peterlin viscoelastic model, a specific decoupling scheme for the continuous-time case is developed. We prove the convergence of this decoupling scheme. Then, for the discrete-time case, a first-order scheme and its iterative error analysis are provided. Numerical experiments validate the theoretical convergence of the algorithm and its dependence on relative parameters.