An electronic control unit executing an algorithm so as to operate an exhaust purification system of an engine. The algorithm (1) commences a regeneration treatment by causing an amount of fuel supplied to a combustion process of the engine to increase so as to change an air-fuel ratio of exhaust gas of the engine from a first lean air-fuel ratio to a set rich air-fuel ratio and (2) causes uncombusted fuel to be supplied to a NOx catalyst device during at least one of: a first period in which an air-fuel ratio of the exhaust gas within the NOx catalyst device changes from the first lean air-fuel ratio to the set rich air-fuel ratio when the regeneration treatment is started; and a second period after an air-fuel ratio of the exhaust gas within the NOx catalyst device becomes a ratio indicating completion of the regeneration treatment.