We reported a narrow-linewidth Brillouin fiber swept laser in a half-open ring cavity based on Brillouin gain medium of a 100-m high nonlinearity fiber (HNLF) as well as distributed Rayleigh random feedback from 1-km single mode fiber (SMF). Thanks to Rayleigh-assistant random lasing resonance, the proposed Stokes laser realizes SLM frequency sweeping radiation as a swept Brillouin pump laser source launched into the random laser cavity. The static linewidth and frequency noise of the Brillouin fiber swept laser is also characterized, which exhibits a linewidth narrowing by over 100 times as well as ~18 dB frequency noise (FN) suppression compared with the pump laser. Results show that the output power and flatness of the Stokes laser were dependent on the sweeping speed of the pump. Demodulated by an unbalanced Mach-Zender interferometer (UMZI), a continuous sweeping range of ~126.33 MHz was obtained during an observation time window of 100 us.