An exhaust catalyst management apparatus to be applied to a vehicle includes a differential pressure sensor and one or more processors. The differential pressure sensor detects an accumulation state of an exhaust-containing substance in a catalyst device. The one or more processors are coupled to the differential pressure sensor and a vehicle outside sensor. The one or more processors determine whether or not travel environment is suited to carrying out an output control for removal, based on a result of the detection of surroundings of the vehicle by a vehicle outside sensor. Upon determining that the travel environment is suited to carrying out the output control for removal, the one or more processors carry out the output control for removal, while the vehicle is traveling, in accordance with a result of the determination of the accumulation state of the exhaust-containing substance in the catalyst device.