Optical spectroscopic sensors are powerful tools for analysing gas mixtures in industrial and scientific applications. Whilst highly sensitive spectrometers tend to have a large footprint, miniaturized optical devices usually lack sensitivity or wideband spectroscopic coverage. By employing a widely tunable, passively stable fiber Fabry-Perot cavity (FFPC), we demonstrate an absorption spectroscopic device that continuously samples over several tens of terahertz. Both broadband scans using cavity mode width spectroscopy to identify the spectral fingerprints of analytes and a fast, low-noise scan method for single absorption features to determine concentrations are exemplary demonstrated for the oxygen A-band. The novel scan method uses an injected modulation signal in a Pound-Drever-Hall feedback loop together with a lock-in measurement to reject noise at other frequencies. The FFPC-based approach provides a directly fiber coupled, extremely miniaturized, light-weight and robust platform for analyzing small analyte volumes that can straightforwardly be extended to sensing at different wavelength ranges, liquid analytes and other spectroscopic techniques with only little adjustments of the device platform.