The first observation of the decays Λ_{b}^{0}→χ_{c1}pK^{-} and Λ_{b}^{0}→χ_{c2}pK^{-} is reported using a data sample corresponding to an integrated luminosity of 3.0 fb^{-1}, collected by the LHCb experiment in pp collisions at center-of-mass energies of 7 and 8 TeV. The following ratios of branching fractions are measured: B(Λ_{b}^{0}→χ_{c1}pK^{-})/B(Λ_{b}^{0}→J/ψpK^{-})=0.242±0.014±0.013±0.009,B(Λ_{b}^{0}→χ_{c2}pK^{-})/B(Λ_{b}^{0}→J/ψpK^{-})=0.248±0.020±0.014±0.009,B(Λ_{b}^{0}→χ_{c2}pK^{-})/B(Λ_{b}^{0}→χ_{c1}pK^{-})=1.02±0.10±0.02±0.05,where the first uncertainty is statistical, the second systematic, and the third due to the uncertainty on the branching fractions of the χ_{c1}→J/ψγ and χ_{c2}→J/ψγ decays. Using both decay modes, the mass of the Λ_{b}^{0} baryon is also measured to be m_{Λ_{b}^{0}}=5619.44±0.28±0.26 MeV/c^{2}, where the first and second uncertainties are statistical and systematic, respectively.