A fiber-optic micro-electromechanical systems (MEMS) extrinsic Fabry-Perot interferometer (EFPI) pressure sensor exploiting microwave photonics filtering technique is firstly proposed and experimentally demonstrated. A single-bandpass microwave photonic filter (MPF) which mainly consists of a spectrum-sliced light source, a pressurized EFPI, a phase modulator (PM) and a length of dispersion compensating fiber (DCF) is demonstrated. The frequency response of the filter with respect to the pressure is studied. By detecting the resonance frequency shifts of the MPF, the pressure can be determined. The theoretical and experimental results show that the proposed EFPI pressure sensor has a higher resolution and higher speed than traditional methods based on optical spectrum analysis. The sensitivity of the sensor is measured to be as high as 86 MHz/MPa in the range of 0-4MPa.