Using e^{+}e^{-} annihilation data corresponding to an integrated luminosity of 3.19 fb^{-1} collected at a center-of-mass energy of 4.178 GeV with the BESIII detector, we measure the absolute branching fractions B_{D_{s}^{+}→ηe^{+}ν_{e}}=(2.323±0.063_{stat}±0.063_{syst})% and B_{D_{s}^{+}→η^{'}e^{+}ν_{e}}=(0.824±0.073_{stat}±0.027_{syst})% via a tagged analysis technique, where one D_{s} is fully reconstructed in a hadronic mode. Combining these measurements with previous BESIII measurements of B_{D^{+}→η^{(')}e^{+}ν_{e}}, the η-η^{'} mixing angle in the quark flavor basis is determined to be ϕ_{P}=(40.1±2.1_{stat}±0.7_{syst})°. From the first measurements of the dynamics of D_{s}^{+}→η^{(')}e^{+}ν_{e} decays, the products of the hadronic form factors f_{+}^{η^{(')}}(0) and the Cabibbo-Kobayashi-Maskawa matrix element |V_{cs}| are determined with different form factor parametrizations. For the two-parameter series expansion, the results are f_{+}^{η}(0)|V_{cs}|=0.4455±0.0053_{stat}±0.0044_{syst} and f_{+}^{η^{'}}(0)|V_{cs}|=0.477±0.049_{stat}±0.011_{syst}.