This paper investigates the problem of robust stability for a class of switched interconnected systems characterized by unstable modes, time-varying state delays, and switching uncertainties. Switching uncertainties cause variations in both the nominal switching sequence and the prescribed switching instants, which significantly impact system stability. To address this, two novel concepts—namely, the generalized nominal switching signal and the composite switching signal—are introduced under switching uncertainty disturbances. Additionally, a new index, termed the generalized mode changing rate, is proposed. Building on these concepts and index, the idea of the vector Lyapunov function and average dwell time method are further incorporated to derive sufficient conditions for ensuring the robust exponential stability of the system. Finally, a numerical example is provided to demonstrate the effectiveness and validity of the proposed theory.