An information processing apparatus that is capable of preventing the overwriting of an incoming packet without disconnecting the network connection. The information processing apparatus receives incoming information via a network, stores it into a storage area, and executes a response process thereto. A response processing unit executes the response process to the incoming information. A first change unit switches so that the incoming information is stored into a second storage area after the incoming information, which is a factor to shift to a power mode from a power saving mode, is stored into a first storage area. A second change unit switches so that the incoming information is stored into the first storage area after the response processing unit executes the response process to the incoming information stored in the first storage area.