Using a data sample corresponding to an integrated luminosity of 11.3 fb^{-1} collected at center-of-mass energies from 4.23 to 4.70 GeV with the BESIII detector, we measure the product of the e^{+}e^{-}→π^{+}π^{-}ψ_{2}(3823) cross section and the branching fraction B[ψ_{2}(3823)→γχ_{c1}]. For the first time, resonance structure is observed in the cross section line shape of e^{+}e^{-}→π^{+}π^{-}ψ_{2}(3823) with significances exceeding 5σ. A fit to data with two coherent Breit-Wigner resonances modeling the sqrt[s]-dependent cross section yields M(R_{1})=4406.9±17.2±4.5 MeV/c^{2}, Γ(R_{1})=128.1±37.2±2.3 MeV, and M(R_{2})=4647.9±8.6±0.8 MeV/c^{2}, Γ(R_{2})=33.1±18.6±4.1 MeV. Though weakly disfavored by the data, a single resonance with M(R)=4417.5±26.2±3.5 MeV/c^{2}, Γ(R)=245±48±13 MeV is also possible to interpret data. This observation deepens our understanding of the nature of the vector charmoniumlike states. The mass of the ψ_{2}(3823) state is measured as (3823.12±0.43±0.13) MeV/c^{2}, which is the most precise measurement to date.