A high-sensitivity pressure sensor based on wavelength detection technology of fiber Bragg grating (FBG) is proposed in this paper. The FBG pressure sensor can be used to measure downhole pressure of oil and gas wells. An eccentric-pushrod structure is used to improve the sensitivity of the sensor. The effect of adhesive on elastic hysteresis of the sensor is analyzed, and the adhesive with large shear modulus is used to reduce the hysteresis of the sensor. Numerical and experimental results show that this sensor possesses good linearity and repeatability, as well as little elastic hysteresis. The sensitivity of the sensor can reach 230.9 pm/MPa when the measured pressure is in the range of 0 to 20 MPa.