An augmented Lyapunov-Krasovskii functional (LKF) approach is presented to derive sufficient conditions for the existence, uniqueness, and globally exponential stability of the equilibrium point of a class of cellular neural networks with time-varying delays. By dividing the variation interval of the time delay into several subintervals with equal length, a novel vector LKF is introduced and new conditions are obtained based on the homeomorphism mapping principle, free-weighting matrix method, and linear matrix inequality techniques. Since the criteria are involving derivative information of both state and delay, the obtained results are less conservative than some previous ones. Two examples are also given to show the effectiveness of the presented criteria.