An inverter control apparatus and a control method thereof are provided. The inverter control apparatus and a control method thereof stably operate a three-phase motor using a capacitor having a small capacitance for a DC link. The inverter control apparatus includes a current sensor to sense an output current of the inverter, a voltage sensor to sense a DC-link voltage of the inverter, and a controller to generate an average of a periodically varying rotor based q-axis current boundary value based on the output current and the DC-link voltage to generate a current reference on the basis of the average of the rotor based q-axis current boundary value, and to drive a three-phase motor based on the current reference. Stabilized variable speed control of a motor by using a small-capacitance capacitor for a DC link of an inverter is performed and reliability of an inverter circuit improved.