The Born cross sections of the process e^{+}e^{-}→D^{*0}D^{*-}π^{+} at center-of-mass energies from 4.189 to 4.951 GeV are measured for the first time. The data samples used correspond to an integrated luminosity of 17.9 fb^{-1} and were collected by the BESIII detector operating at the BEPCII storage ring. Three enhancements around 4.20, 4.47, and 4.67 GeV are visible. The resonances have masses of 4209.6±4.7±5.9 MeV/c^{2}, 4469.1±26.2±3.6 MeV/c^{2}, and 4675.3±29.5±3.5 MeV/c^{2} and widths of 81.6±17.8±9.0 MeV, 246.3±36.7±9.4 MeV, and 218.3±72.9±9.3 MeV, respectively, where the first uncertainties are statistical and the second systematic. The first and third resonances are consistent with the ψ(4230) and ψ(4660) states, respectively, while the second one is compatible with the ψ(4500) observed in the e^{+}e^{-}→K^{+}K^{-}J/ψ process. These three charmoniumlike ψ states are observed in the e^{+}e^{-}→D^{*0}D^{*-}π^{+} process for the first time.