An installation structure of an elastic gasket that is attached and installed as a sealing member between an upper surface of a window frame and a glass supporting insulation bracket installed on an upper or lower portion of a glass panel constituting a moving window in a sliding window system having a two-sided supporting frame window, wherein the installation structure of the elastic gasket comprises: the glass supporting insulation bracket formed and combined with a synthetic material to support the glass panel at the upper or lower portion of the glass panel constituting the moving window and to insulate between an inside and outside of the moving window, whereby the glass supporting insulation bracket is provided with a rotation groove at an inner corner between a side surface and a bottom surface; a rotation fastener of an aluminum material is inserted and installed in the rotation groove, wherein the rotation fastener comprises a rotation end portion that is rotatably fitted into the rotation groove, and a fitting groove that is formed on an outer surface extending upward beyond the side surface of the glass supporting insulation bracket; and an elastic gasket (2s2) provided by fitting a plastic sliding cap (2s2c) on an outer elastic end (2s2e) of the gasket so as to achieve a sliding seal between the side surface of the glass supporting insulation bracket and the upper surface of the window frame during sliding movement of the moving window, wherein the elastic gasket (2s2) is formed of a rubber material and is arranged for an upper contact end (2s2b) of the gasket to support the front or back surface of the glass panel so that the fixing end (2s2a) of the gasket is fitted into the fitting groove of the rotation fastener, and the outer elastic end (2s2e) of the gasket is engaged in such a manner as to cover the upper surface of the sash frame made of aluminum in a downwardly contacting manner. the glass supporting insulation bracket formed and combined with a synthetic material to support the glass panel at the upper or lower portion of the glass panel constituting the moving window and to insulate between an inside and outside of the moving window, whereby the glass supporting insulation bracket is provided with a rotation groove at an inner corner between a side surface and a bottom surface; a rotation fastener of an aluminum material is inserted and installed in the rotation groove, wherein the rotation fastener comprises a rotation end portion that is rotatably fitted into the rotation groove, and a fitting groove that is formed on an outer surface extending upward beyond the side surface of the glass supporting insulation bracket; and an elastic gasket (2s2) provided by fitting a plastic sliding cap (2s2c) on an outer elastic end (2s2e) of the gasket so as to achieve a sliding seal between the side surface of the glass supporting insulation bracket and the upper surface of the window frame during sliding movement of the moving window, wherein the elastic gasket (2s2) is formed of a rubber material and is arranged for an upper contact end (2s2b) of the gasket to support the front or back surface of the glass panel so that the fixing end (2s2a) of the gasket is fitted into the fitting groove of the rotation fastener, and the outer elastic end (2s2e) of the gasket is engaged in such a manner as to cover the upper surface of the sash frame made of aluminum in a downwardly contacting manner.