To meet the requirements of high sensitivity, resolution, and reliability in pressure measurements, in this paper, we proposed a high-precision optical fiber pressure sensor with a diaphragm-type Fabry-Perot (F-P) cavity based on the demodulation principle of frequency-modulated continuous wave (FMCW) laser interference. Firstly, a pressure measurement model based on this principle was deduced. Secondly, a pressure measuring device was developed and subsequently the sealed cavity formed by the SUS631 stainless steel diaphragm was continuously inflated through an air pump to collect the characteristic curve of the central deformation of the SUS631 stainless steel diaphragm (which is the amount of change in the F-P cavity length) with the air pressure when the air pressure changed in the range of 0--600 kPa. Furthermore, the cavity change of F-P cavity was measured at 25 ℃ and different pressures and the causes inducing the drift of the cavity change were discussed. Finally, the resolution of pressure measurements without the system error was acquired. The results show that the FMCW pressure sensor has a sensitivity of 286.55 nm/kPa and a resolution of 0.287 nm/Pa and its random measurement error displays a normal distribution. The above analyses verify that the proposed FMCW optical fiber pressure sensor can perform measurements with high sensitivity, resolution, and stability.