Raman spectroscopy is an extremely powerful laser-based method for characterizing materials based on their unique inelastic scattering spectrum. Ultimately, the power of the technique is limited by the resolution of the spectrometer. Here we introduce a new method for achieving Super-Spectral-Resolution Raman Spectroscopy (SSR-RS), by angle-tuning a Fabry-Pérot (F-P) etalon filter that we incorporated in a micro-Raman setup. A monolithically coated F-P etalon structure, only 1.686 mm in thickness, was mounted onto an angle-tunable motorized stage, and Raman spectra were automatically acquired for many different angles of the etalon. Using a low-resolution grating of 150 g/mm by itself, without the F-P etalon, we obtained a best-case regular Raman spectral linewidth of 44 cm-1 for the characteristic Raman peak from a diamond sample. When we applied the SSR-RS technique to diamond, we obtained a super-spectral resolution peak that was 27x narrower, namely 1.63 cm-1, and a Raman shift of 1331.3 cm-1. To baseline SSR-RS, we applied the super-spectral-resolution method to extract the linewidth and peak wavelength of the laser excitation itself and obtained a laser linewidth of better than 0.014 cm-1, with a laser wavelength centered at 531.962 nm, close to the stated wavelength of 532 nm. This extracted laser linewidth is 3300x times narrower compared to its measured linewidth of 46 cm-1. Thus, our work suggests that SSR-RS can be very generally applied to greatly improve the resolution and precision of Raman instrumentation, and potentially lower the cost of obtaining high-resolution Raman spectroscopic capabilities.