An approach to the interrogation of a linearly chirped fiber Bragg grating (LCFBG) sensor using a linearly frequency-modulated (or chirped) optical waveform (LFMOW) with a high resolution is proposed and experimentally demonstrated. An LFMOW is generated at a laser diode through linear frequency modulation. The generated LFMOW is then launched into an LCFBG pair consisting of two identical LCFBGs, with one serving as a sensing LCFBG and the other as a reference LCFBG. The reflection of the LFMOW from the two LCFBGs would lead to two time delayed LFMOWs. By beating the LFMOWs at a photodetector, a microwave signal with a beat frequency that is proportional to the time delay difference between the two reflected LFMOWs is generated. By measuring the frequency change of the beat signal, the strain applied to the sensing LCFBG is estimated. The proposed approach is experimentally evaluated. An LCFBG sensor with a resolution of 0.25 με is experimentally demonstrated.