Raman lasers are well known for wavelength shifting and beam clean up, but less so for linewidth narrowing. Recent advances in design and material choice have enabled stable single frequency operation and the opportunity to investigate their narrow linewidth properties. Here, we use modeling and experiments to investigate linewidth and frequency noise as a function of pump noise. The model reveals that the picosecond-scale phonon dephasing time, typical of most Raman materials, suppresses pump frequency noise by factors up to 108 at Fourier frequencies up to 1011 Hz. To support the model, frequency noise spectra were measured for the input and output beams of a singly-resonant Raman laser operating at 1178 nm, revealing noise suppression by more than 104 at offset frequencies above 1 MHz. The results show that Raman lasers offer orders-of-magnitude greater linewidth-narrowing than Brillouin lasers, even for cavities of moderate Q factor. This makes them a promising route toward ultra-narrow linewidth lasers across the spectrum with adaptability to a wide range of pumps.