An enhanced distributed acoustic sensing (DAS) is proposed based on an extended Mach-Zehnder interferometer utilizing quantum weak measurement. The acoustic signals are encoded as the relative phase of the polarized light in several channels by fibers between optical fiber Bragg gratings (FBGs). With appropriate preselection and postselection, the acoustic signal can be extracted by performing the Fourier transform on the contrast ratio of the detected light intensity. Theoretical analysis shows that the scheme can achieve a spatial resolution of 1 m, and the system noise can be decreased by three orders of magnitude compared to classical distributed acoustic sensing. Moreover, this scheme might have potential application in long-distance acoustic source localization.