This paper is concerned with the delay-dependent stability of systems with time-varying delays. The novelty relies on the use of the second-order Bessel–Legendre integral inequality which is less conservative than the Jensen and Wirtinger-based inequalities. Unlike similar contributions, the features of this inequality are fully integrated into the construction of augmented Lyapunov–Krasovskii functionals leading to novel stability criteria expressed in terms of linear matrix inequalities. The stability condition is tested on some classical numerical examples illustrating the efficiency of the proposed method.