We describe short-term swing dynamics leading to cascading failure in an interconnected power system. The notion of Coherent Swing Instability (CSI) was recently proposed in which most of synchronous machines in a power system coherently lose synchronism with the rest of the system after being subjected to a local and finite disturbance. CSI is an escape phenomenon from resonance in the nonlinear mode that governs the collective motion of machines. For a network of weakly interconnected power systems, we show that CSI happens for all of the systems in a cascade manner and gives the mechanism of cascading failure in the network, that is, the sequence of escape phenomena that can occur in a class of high-dimensional dynamical systems.