We report the first observation of the Cabibbo-suppressed charm baryon decay Ξ_{c}^{+}→pK^{-}π^{+}. We observe 150±22±5 events for the signal. The data were accumulated using the SELEX spectrometer during the 1996-1997 fixed target run at Fermilab, chiefly from a 600 GeV/c Σ^{-} beam. The branching fractions of the decay relative to the Cabibbo-favored Ξ_{c}^{+}→Σ^{+}K^{-}π^{+} and Ξ_{c}^{+}→Ξ^{-}π^{+}π^{+} are measured to be B(Ξ_{c}^{+}→pK^{-}π^{+})/B(Ξ_{c}^{+}→Σ^{+}K^{-}π^{+})=0.22±0.06±0.03 and B(Ξ_{c}^{+}→pK^{-}π^{+})/B(Ξ_{c}^{+}→Ξ^{-}π^{+}π^{+})=0.20±0.04±0.02, respectively.