This paper is concerned with stability of a linear system with a time-varying delay. The direct Lyapunov method is a powerful tool for studying stability of the system. Note that a tighter estimation on the time derivative of some Lyapunov-Krasovskii functional usually leads to a less conservative stability criterion. First, by introducing an auxiliary vector-valued function, this paper proposes a novel integral inequality, which can provide a tighter estimation on the integral term appearing in the time derivative of the Lyapunov-Krasovskii functional. Second, by introducing an augmented Lyapunov-Krasovskii functional, this novel integral inequality is employed to derive a stability criterion for the system with a time-varying delay. Finally, it is shown through a well-studied numerical example that the proposed stability criterion outperforms those using the IQC approach, the quadratic separation approach, the Wirtinger-based integral inequality approach and the free-matrix-based integral inequality approach.