The first observation of the suppressed semileptonic B_{s}^{0}→K^{-}μ^{+}ν_{μ} decay is reported. Using a data sample recorded in pp collisions in 2012 with the LHCb detector, corresponding to an integrated luminosity of 2 fb^{-1}, the branching fraction B(B_{s}^{0}→K^{-}μ^{+}ν_{μ}) is measured to be [1.06±0.05(stat)±0.08(syst)]×10^{-4}, where the first uncertainty is statistical and the second one represents the combined systematic uncertainties. The decay B_{s}^{0}→D_{s}^{-}μ^{+}ν_{μ}, where D_{s}^{-} is reconstructed in the final state K^{+}K^{-}π^{-}, is used as a normalization channel to minimize the experimental systematic uncertainty. Theoretical calculations on the form factors of the B_{s}^{0}→K^{-} and B_{s}^{0}→D_{s}^{-} transitions are employed to determine the ratio of the Cabibbo-Kobayashi-Maskawa matrix elements |V_{ub}|/|V_{cb}| at low and high B_{s}^{0}→K^{-} momentum transfer.