Magnetic reconnection is a process whereby magnetic field lines in different directions reconnect with each other, resulting in the rearrangement of magnetic field topology together with the conversion of magnetic field energy into the kinetic energy (K.E.) of energetic particles. This process occurs in magnetized astronomical plasmas, such as those in the solar corona, Earths magnetosphere, and active galactic nuclei, and accounts for various phenomena, such as solar flares, energetic particle acceleration, and powering of photon emission. In the present study, we report the experimental demonstration of magnetic reconnection under relativistic electron magnetization situation, along with the observation of power-law distributed outflow in both electron and proton energy spectra. Through irradiation of an intense laser on a micro-coil, relativistically magnetized plasma was produced and magnetic reconnection was performed with maximum magnetic field 3 kT. In the downstream outflow direction, the non-thermal component is observed in the high-energy part of both electron and proton spectra, with a significantly harder power-law slope of the electron spectrum (p = 1.535 +/- 0.015) that is similar to the electron injection model proposed to explain a hard emission tail of Cygnus X-1, a galactic X-ray source with the same order of magnetization. The obtained result showed experimentally that the magnetization condition in the emitting region of a galactic X-ray source is sufficient to build a hard electron population through magnetic reconnection.