A numerical approach to approximate the nonlinear Burgers’ equation solution is presented in this article. By temporally discretizing the problem using the Crank-Nicolson scheme, we present a discrete formulation along with approximate solutions at each time step. Next, we express these approximations in terms of the Chelyshkov polynomial basis. Then, Newton’s method is utilized to solve the resulting nonlinear system of equations to find the values of unknowns. Three numerical examples are provided to assess the accuracy and efficiency of the suggested method. The proposed approach demonstrates more accurate numerical results than those existing in the literature.