We report a possible discovery of a relic star cluster in the Sextans dwarf spheroidal galaxy. Using the hk index (≡(Ca-b-(b-y)) as a photometric metallicity indicator, we discriminate the metal-poor and metal-rich stars in the galaxy and find unexpected number density excess of metal-poor stars located 7.7' (~190 pc in projected distance) away from the known galactic center. The V-I color-magnitude diagram (CMD) for stars around the density excess reveals that both the main sequence and the giant branch are considerably narrower and redder than the bulk of field stars in Sextans. Our stellar population models show (a) that the narrow CMD is best reproduced by a simple stellar population with an age of ~13 Gyr and [Fe/H] of ~ -2.3 dex, and (b) that the redder V-I color of the hk-weak population is explained only if it is ~2 Gyr older than the field stars. The results lead us to conclude that the off-centered density peak is likely associated with an old, metal-poor globular cluster. The larger spatial extent (>80 pc in radius) and the smaller number of stars (~1000) than typical globular clusters point to a star cluster that is in the process of dissolution. The finding serves as the first detection of a surviving star cluster in Sextans, supporting previous suggestions of the presence of star cluster remnants in the galaxy. If confirmed, the survival of a relic star cluster until now implies a cored dark matter halo profile for this dwarf galaxy.