A low-noise Brillouin random fiber laser (BRFL) with stabilized lasing output and narrow linewidth is proposed and demonstrated experimentally. Stimulated Brillouin scattering and Rayleigh scattering are exploited to work as the gain and random distributed feedback mechanisms, respectively. A simple unpumped Erbium-doped fiber (EDF) loop structure which consists of a section of EDF and an optical coupler is introduced into the BRFL cavity, forming a saturable absorption ring (SAR). When light with proper intensity is incident into the SAR, a dynamic fiber grating (DFG) can be built up to work as a narrow-bandwidth filter with the ability of auto-tracking the center frequency. The SAR can effectively filter out dense random longitudinal modes and suppress multi-mode resonances and frequency jitters by restricting the mode-competition-induced mode hopping within a limited frequency range of ∼ 1 MHz and auto-tracking the center frequency of light in the cavity, simultaneously. Thus, the output of BRFL with SAR is much more stabilized than existing free-running BRFLs. In the experiment, a stable lasing output with an ultra-narrow linewidth (less than2kHz) is obtained from the proposed BRFL. The measured relative intensity noise and frequency noise demonstrate that the noise level of the proposed laser is significantly lower than that of the free-running BRFLs. This proposed technique not only provides a solution to the dilemma of previously reported BRFLs with severe lasing instabilities, but also makes a step towards the development of practical and reliable BRFLs. This high-quality laser source with advantages of high stability, low cost, low threshold power, high coherence, narrow linewidth, and low noise can find useful applications in coherent optical detection, high-resolution spectrometers, microwave photonics, and optical sensing fields.