The process e^{+}e^{-}→D_{s}^{*+}D_{s}^{*-} is studied with a semi-inclusive method using data samples at center-of-mass energies from threshold to 4.95 GeV collected with the BESIII detector operating at the Beijing Electron Positron Collider. The Born cross sections of the process are measured for the first time with high precision in this energy region. Two resonance structures are observed in the energy-dependent cross sections around 4.2 and 4.4 GeV. By fitting the cross sections with a coherent sum of three Breit-Wigner amplitudes and one phase-space amplitude, the two significant structures are assigned masses of (4186.8±8.7±30) and (4414.6±3.4±6.1) MeV/c^{2}, widths of (55±15±53) and (122.5±7.5±8.1) MeV, where the first errors are statistical and the second ones are systematic. The inclusion of a third Breit-Wigner amplitude is necessary to describe a structure around 4.79 GeV.