In this paper, we investigate the problem of exponential stability and passivity analysis of a class of switched systems with interval time-varying delays and nonlinear perturbations. By constructing an improved Lyapunov–Krasovskii functional combining with novel refined Jensen-based inequalities, some improved sufficient conditions for exponential stability are proposed for a class of switching signals with average dwell time. Moreover, a new sufficient condition for passivity analysis of switched continuous-time systems with an interval time-varying delay is also derived. These conditions are delay dependent and are given in the form of linear matrix inequalities, which therefore can be efficiently solved by existing convex algorithms. Lastly, four examples are provided to demonstrate the effectiveness of our results.