Mainly addressed in this paper is the stability problem of continuous-time switched cascade nonlinear systems with time-varying delays. A robust convergence property is proved first: If a nominal switched nonlinear system with delays is asymptotically stable, then trajectories of corresponding perturbed system asymptotically approach origin provided that the perturbation can be upper bounded by a function exponentially decaying to zero. Applying this property and assuming that a cascade system consists of two separate systems, it is shown that a switched cascade nonlinear system is asymptotically stable if one separate system is exponentially stable and the other one is asymptotically stable. Since the considered switching signals have a uniform property and thus include most switching signals frequently encountered, our results are valid for a wide range of switched cascade systems.