The first observation of the decays Lambda(b)(0) -> chi(c1)pK(-) and Lambda(0)(b) -> chi(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(Lambda(0)(b) -> chi(c1)pK(-))/B(Lambda(0)(b) -> J/psi pK(-)) = 0.242 +/- 0.014 +/- 0.013 +/- 0.009, B(Lambda(0)(b) -> chi(c2)pK(-))/B(Lambda(0)(b) -> J/psi pK(-)) = 0.248 +/- 0.020 +/- 0.014 +/- 0.009, B(Lambda(0)(b) -> chi(c2)pK(-))/B(Lambda(0)(b) -> chi(c1)pK(-)) = 1.02 +/- 0.010 +/- 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 x(c1) -> J/psi gamma and chi(c2) -> J/psi gamma decays. Using both decay modes, the mass of the Ab baryon is also measured to be m(Lambda b0) = 5619.44 +/- 0.28 +/- 0.26 MeV/c(2), where the first and second uncertainties are statistical and systematic, respectively.