The Lyapunov-Krasovskii functional (LKF) method is an effective tool to obtain delay-dependent stability criteria of time-varying delay systems. However, in the method, the requirement that there must exist an appropriate LKF for a system with a delay varying in a delay interval is too strict and may lead to considerable conservativeness. In order to get less conservative stability criteria, a novel stability analysis method called a delay-mode-based LKF (DMBLKF) method is first presented for nonlinear delay systems. Compared with the LKF method, the main advantage of the DMBLKF method lies in relaxing the aforementioned requirement and leading to a less conservative criterion in stability analysis. Then, with the help of the novel method, new stability criteria of linear delay systems are derived. Finally, the effectiveness of the proposed method and the obtained criteria is illustrated through two numerical examples.