This article reports on a search for dark matter pair production in association with a Higgs boson decaying to a pair of bottom quarks, using data from 20.3 fb-1 of pp collisions at a center-of-mass energy of 8 TeV collected by the ATLAS detector at the LHC. The decay of the Higgs boson is reconstructed as a high-momentum bb system with either a pair of small-radius jets, or a single large-radius jet with substructure. The observed data are found to be consistent with the expected Standard Model backgrounds. Model-independent upper limits are placed on the visible cross sections for events with a Higgs boson decaying into bb and large missing transverse momentum with thresholds ranging from 150 to 400 GeV. Results are interpreted using a simplified model with a Z′ gauge boson decaying into different Higgs bosons predicted in a two-Higgs-doublet model, of which the heavy pseudoscalar Higgs decays into a pair of dark matter particles. Exclusion limits are also presented for the mass scales of various effective field theory operators that describe the interaction between dark matter particles and the Higgs boson.