A novel high-sensitivity fiber Bragg grating (FBG) strain sensor is reported, to the best of our knowledge. The sensitivity of the sensor is improved by fixing the FBG on an elastic substrate with a sensitization function. The sensitization principle of the designed sensor is introduced, and the mathematical model of the sensor is established. In the static and dynamic experiments of the sensor, the effect of adhesive between the sensor and the measured structure on the sensitivity of the FBG strain sensor is experimentally investigated. The experimental results show that the adhesive with high shear strength is beneficial to the realization of a high-sensitivity sensor. The sensor fixed with planting bar glue can achieve a sensitivity of 9.42 pm/µε, a repeatability error of 4.79%, and a hysteresis error of 3.36%, which is consistent with theoretical and simulation results. The designed high-sensitivity strain sensor has a simple structure, small size, and convenient installation, so it has a good application prospect in micro-strain monitoring.