The exponential stability of zero solutions of a class of impulsive functional differential equations in an unbounded region is studied. The solution of the system is derived by using the Fourier transform method, and the Cauchy matrix of the linear system is estimated by using the inequality reduction technique. Finally, a sufficient condition for the global exponential stability of the zero solution of the nonlinear system is given by the established differential inequalities and the assumed conditions. Under the assumption that the nonlinear system satisfies the given conditions, the zero solution is globally exponentially stable. The results of this study extend the related works in the existing literature.