Using 2.93 fb^{-1} of e^{+}e^{-} collision data taken at a center-of-mass energy of 3.773 GeV with the BESIII detector, we report the first measurements of the absolute branching fractions of 14 hadronic D^{0(+)} decays to exclusive final states with an η, e.g., D^{0}→K^{-}π^{+}η, K_{S}^{0}π^{0}η, K^{+}K^{-}η, K_{S}^{0}K_{S}^{0}η, K^{-}π^{+}π^{0}η, K_{S}^{0}π^{+}π^{-}η, K_{S}^{0}π^{0}π^{0}η, and π^{+}π^{-}π^{0}η; D^{+}→K_{S}^{0}π^{+}η, K_{S}^{0}K^{+}η, K^{-}π^{+}π^{+}η, K_{S}^{0}π^{+}π^{0}η, π^{+}π^{+}π^{-}η, and π^{+}π^{0}π^{0}η. Among these decays, the D^{0}→K^{-}π^{+}η and D^{+}→K_{S}^{0}π^{+}η decays have the largest branching fractions, which are B(D^{0}→K^{-}π^{+}η)=(1.853±0.025_{stat}±0.031_{syst})% and B(D^{+}→K_{S}^{0}π^{+}η)=(1.309±0.037_{stat}±0.031_{syst})%, respectively. The charge-parity asymmetries for the six decays with highest event yields are determined, and no statistically significant charge-parity violation is found.