The first observation of the decays Λ0b → χc1 p K− and Λ0b → χc2 p K− 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(Λ0b → χc1 p K−)/B(Λ0b → J/ψ p K−) = 0.242 ± 0.014 ± 0.013 ± 0.009, B(Λ0b → χc2 p K−)/B(Λ0b → J/ψ p K−) = 0.248 ± 0.020 ± 0.014 ± 0.009, B(Λ0b → χc2 p K−)/B(Λ0b → χc1 p K−) = 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 χc1 → J/ψ γ and χc2 → J/ψ γ decays. Using both decay modes, the mass of the Λ0b baryon is also measu...