In this article, a new approach—namely, the extended Parker–Sochacki method (EPSM)—is presented for solving the Michaelis–Menten nonlinear enzymatic reaction model. The Parker–Sochacki method (PSM) is combined with a new resummation method called the Sumudu–Padé resummation method to obtain approximate analytical solutions for the model. The obtained solutions by the proposed approach are compared with the solutions of PSM and the Runge–Kutta numerical method (RKM). The comparison proves the practicality, efficiency, and correctness of the presented approach. It serves as a basis for solving other nonlinear biochemical reaction models in the future.