Solving problems in practice often results in a system of nonlinear equations with a large number of equations and unknowns. Finding the exact solution to this class of equations is very difficult and almost impossible. Recently, with the development of technology, many methods and algorithms have been proposed to approximate the class of these systems of equations. Especially the third-order Newton–Krylov method has solved quite well this class of systems of equations with the third degree of convergence. In this paper, we present a new improvement of the third-order Newton-Krylov method with a quaternary convergence rate and prove the convergence of the iterative formula. In addition, the paper also presents an experimental result to demonstrate the convergence speed of the method.