A control device for a drive device of a vehicle which is provided with an engine, a supercharger to raise a pressure of intake air introduced in said engine, an accelerating member manually operable by an operator of the vehicle to accelerate the vehicle, and an automatic transmission constituting a portion of a power transmitting path between said engine and drive wheels, and wherein said engine is placed in a first operating state in which said supercharger is operated, or a second operating state in which an operation of said supercharger is restricted as compared with that in said first operating state, said control device comprising: a speed ratio selection control portion configured to implement a speed ratio selection control to change a method of selecting a speed ratio of said automatic transmission depending upon a rate of increase of an operation amount of said accelerating member, where said engine will be placed in said first operating state if said automatic transmission is shifted to select a first speed ratio value according to a predetermined shifting map while said engine is placed in said second operating state, said speed ratio selection control portion implementing said speed ratio selection control to shift said automatic transmission to select said first speed ratio value when the rate of increase of the operation amount of said accelerating member is relatively low, and to shift said automatic transmission to select a second speed ratio value at which said engine is kept in said second operating state and which is higher than said first speed ratio value, when the rate of increase of the operation amount of said accelerating member is relatively high, said speed ratio selection control portion implements said speed ratio selection control such that said automatic transmission which has been shifted to select said second speed ratio value is shifted to select said first speed ratio value, said control device being characterized in that said speed ratio selection control portion switching said engine from said second operating state to said first operating state after a moment of completion of a shifting action of said automatic transmission to select said second speed ratio value and before a moment of initiation of a shifting action of the automatic transmission from said second speed ratio value to select said first speed ratio value. a speed ratio selection control portion configured to implement a speed ratio selection control to change a method of selecting a speed ratio of said automatic transmission depending upon a rate of increase of an operation amount of said accelerating member, where said engine will be placed in said first operating state if said automatic transmission is shifted to select a first speed ratio value according to a predetermined shifting map while said engine is placed in said second operating state, said speed ratio selection control portion implementing said speed ratio selection control to shift said automatic transmission to select said first speed ratio value when the rate of increase of the operation amount of said accelerating member is relatively low, and to shift said automatic transmission to select a second speed ratio value at which said engine is kept in said second operating state and which is higher than said first speed ratio value, when the rate of increase of the operation amount of said accelerating member is relatively high, said speed ratio selection control portion implements said speed ratio selection control such that said automatic transmission which has been shifted to select said second speed ratio value is shifted to select said first speed ratio value, said control device being characterized in that said speed ratio selection control portion switching said engine from said second operating state to said first operating state after a moment of completion of a shifting action of said automatic transmission to select said second speed ratio value and before a moment of initiation of a shifting action of the automatic transmission from said second speed ratio value to select said first speed ratio value.