We investigate exponential stability of time-varying delay systems in this article. By considering intermediate polynomials and introducing slack variables, we develop new intermediate polynomial-based weighted functions (IPWFs). Then an exponential stability criterion, in terms of linear matrix inequalities (LMIs), is obtained by choosing an appropriate Lyapunov–Krasovskii functional (LKF) and some novel techniques on IPWFs. Numerical results show that the criterion obtained in this article is less conservative than some previous results.