Achieving cavity-optomechanical strong coupling with high-frequency phonons provides a rich avenue for quantum technology development, including quantum state transfer, memory, and transduction, as well as enabling several fundamental studies of macroscopic phononic degrees of freedom. Reaching such coupling with GHz mechanical modes, however, has proved challenging, with a prominent hindrance being material- and surface-induced optical absorption in many materials. Here, we circumvent these challenges and report the observation of optomechanical strong coupling to a high-frequency (11 GHz) mechanical mode of a fused-silica whispering-gallery microresonator via the electrostrictive Brillouin interaction. Using an optical heterodyne detection scheme, the anti-Stokes light back-scattered from the resonator is measured, and normal-mode splitting and an avoided crossing are observed in the recorded spectra, providing unambiguous signatures of strong coupling. The optomechanical coupling rate reaches values as high as G/2 pi=39 MHz through the use of an auxiliary pump resonance, where the coupling dominates both optical (kappa/2 pi = 3 MHz) and mechanical (gamma(m)/2 pi = 21 MHz) amplitude decay rates. Our findings provide a promising new approach for optical quantum control using light and sound.