The fundamental problem of time-optimal queue control in packet-switched networks is how to adjust source rates in time after network disturbances so that the network queue sizes converge to desired values in the minimum time, while ensuring that always at least one link remains fully utilized in every flow's path. This nonlinear feedback control problem had been solved in a previous paper [Time-Optimal Network Queue Control: The Case of a Single Congested Node] for a single queue in a single congested node and the solution proven robust to queue size and bandwidth estimation errors. In this paper, we generalize that solution to a general network of flows crisscrossing queues, with link delays being arbitrary. The solution, derived for desired queue sizes of 0, turns out to be simple: two computationally simple conditions on the source rate control, viz. QRE-feasibility and maximally utilizing property are sufficient to ensure time-optimality, regardless of the packet scheduling scheme used inside the network nodes.