An apparatus for moving a motor-operated seat rail, including a lead screw provided along an axial direction of a fixed rail, a mounting bracket having opposite ends combined with a moving rail, with the lead screw passing through a central portion of the mounting bracket so that the bracket moves along the lead screw, and a spindle nut engaged with the lead screw through a screw type engagement. A first end of the spindle nut is rotatably held on the mounting bracket so as to rotate relative to the bracket, and a second end passes through the central portion of the mounting bracket and extends along an axial direction of the lead screw, and being configured such that the inner diameter of the second end is able to be elastically reduced so that the inner circumferential surface of the second end comes into close contact with the outer circumferential surface of the lead screw.