Glass substrate cutting apparatuses using lasers are disclosed, where a laser cutting head is moved. A glass substrate cutting apparatus includes two parts for a laser cutting head: heavy laser beam generators fixed to respective ends of a gantry structure moving in parallel along two gantry stages located on either side of a cutting table, and relatively lightweight laser irradiation heads moving horizontally in parallel with the gantry structure. The glass substrate cutting apparatus includes a cutting table for maintaining a glass substrate in a horizontal state; biaxial gantry stages for moving a gantry structure along the cutting table; the gantry structure moving in between an upper part of the biaxial gantry stages; laser beam generators fixed to respective ends of the gantry structure for oscillating the laser; and laser irradiation heads that move horizontally on respective ends of the gantry structure and irradiate the laser upon the glass substrate.