In this Letter, we present and demonstrate a novel technique for distributed measurements in Brillouin optical time-domain analysis based on the use of the nonlinear phase-shift induced by stimulated Brillouin scattering. Employing a Sagnac interferometer (SI), the position-resolved Brillouin phase-shift spectrum (BPS) along the fiber can be obtained, benefiting from the sensitivity to nonreciprocal phase-shifts of the SI scheme. This proposal simplifies the existing methods to retrieve the BPS distribution along an optical fiber since phase modulation, filtering, and high-bandwidth detectors are not required. The fundamentals of the technique are described theoretically and validated through numerical simulations and experimental measurements.