We measure the absolute branching fractions of semileptonic D_s⁺, decays via the e⁺e^- → D_s^*+D_s^*- process using e⁺e^- collision data corresponding to an integrated luminosity of 10.64 fb^-1 collected by the BESIII detector at center-of-mass energies between 4237 and 4.699 GeV. The branching fractions are B(D_s⁺ → ηe⁺ν_e) = (2.35 ± 0.11_stat ± 0.10_syst)%, B(D_s⁺ → η'e⁺ν_e) = (0.82 ± 0.09_stat ± 0.04_syst)%, B(D_s⁺ → Φe⁺ν_e) = (2.21 ± 0.16_stat ± 0.11_syst)%, B(D_s⁺ → f(0)(980)e⁺ν_e), f(0)(980) → π⁺π^-) = (0.15 ± 0.02_stat ± 0.01_syst)%, B(D_s⁺ → K⁰e⁺ν_e) = (0.24 ± 0.04_stat ± 0.01_syst)%, and B(D_s⁺ → K⁰e⁺ν_e) = (0.19 ± 0.03_stat ± 0.01_syst)%. These results are consistent with those measured via the e⁺e^- → D_s^*+/-D_s^-/+ process by BESIII and CLEO. Using two -parameter series expansion, the hadronic transition form factors of D_s⁺ → ηe⁺ν_e, D_s⁺ → η'e⁺ν_e and D_s⁺ → K⁰e⁺ν_e are determined to be f₊^η(0) = 0.442 ± 0.022_stat ± 0.017_syst, f₊^η'(0) = 0.557 ± 0.062_stat ± 0.024_syst, and f₊^{K^0}(0) = 0.677 ± 0.098_stat ± 0.023_syst.