Handheld spectrometers are gaining attention due to their growing market. The customers seek to analyze their own samples with good accuracy and reasonable cost. Therefore, the spectrometer manufacturers miniaturize their products and reduce their cost. However, this leads to decreased spectral resolution and optical throughput rendering the task of the identifying closely spaced spectral lines challenging. In this work, we report the application of the compressive sensing (CS) techniques on a MEMS Fourier Transform Infrared spectrometer, for the sake of resolution enhancement based on the spectrum sparsity. The spectrometer wavelength range is 1300–2500 nm while its core engine is a micromachined scanning Michelson interferometer. The interferometer scanning mirror is driven by a MEMS electrostatic actuator with programmable travel range corresponding to two different resolutions of about 16 nm and 8 nm around 1550 nm. The CS algorithm is applied on filtered white light around a wavelength of 2000 nm fed to the spectrometer using multimode optical fiber and is found to enhance the resolving power down to 3 nm starting from the 22 nm resolution. Then the algorithm is applied on larger number of lines by superposing the spectral lines of a tuneable laser source around 1550 nm. The spacing between the spectral lines is varied and the reconstructed spectra by direct FFT and using the CS technique are compared. The CS technique shows overall better spectral resolution the efficiency of the technique is found to deteriorate as the number of spectral lines increases.