Large-scale cascading failures can be triggered by very few initial failures, leading to severe damages in complex networks. As modern society becomes more and more networked, there is an increasing requirement of security and reliability of complex networks such as infrastructure networks and cyber networks. In order to design networks which are robust to attacks and enhance the security of the existing networks, this paper studies load-dependent cascading failures in random networks consisting of a large but finite number of components. Under a random single-node attack, a framework is developed to quantify the damage at each stage of a cascade. We mainly use probability theory to analyze the cascade process and use simulations to verify our conclusion. In our result, estimations for the fraction of failed nodes are presented to evaluate the time-dependent system damage due to the attack. Furthermore, the analysis reveals a phase transition behavior in the extent of the damage as the load margin grows. That is, the fraction of the damaged components drops from near one to near zero over a slight change in the load margin. The critical value of the load margin and the short interval over which such an abrupt change occurs are derived to characterize the network reaction to small network load variations. Our findings provide design principles for enhancing the network resiliency and provide guidelines for choosing the load margin to avoid a cascade of failures in load-dependent complex networks with practical sizes.