We introduce an enhanced method for the characterization of stimulated Brillouin scattering (SBS) spectra in single-mode fiber that is based on optical single-sideband modulation. This novel technique is shown to provide high-resolution characterization of SBS even under saturation operation in a simple and stable setup in which the spectrum is translated from the optical to the electrical domain, sweeping the frequency of an electrical signal generator. Experimental results are used to demonstrate the performance of the system in measuring the detailed structure of acoustic modes in three types of single-mode fiber.