Nowadays the Krasovskii functional method is still widely used for stability analysis of many delay systems. It means that improving the resultful method is significative for obtaining new stability criteria. Thus, exponential stability analysis for non-switched and switched systems with interval time-varying delays is restudied in this paper. A generalization of the Krasovskii functional method for nonlinear delay systems is proposed. This is achieved by weakening a condition in the Krasovskii functional method. It is proved that the generalized method can be extended to study switched nonlinear systems with delays. How to apply the new method to obtain stability criteria for non-switched and switched linear systems with interval time-varying delays is also shown and three examples are given to illustrate that the conditions of the linear systems are less conservative than some existing criteria.