The spin and parity of the Z_{c}(3900)^{±} state are determined to be J^{P}=1^{+} with a statistical significance larger than 7σ over other quantum numbers in a partial wave analysis of the process e^{+}e^{-}→π^{+}π^{-}J/ψ. We use a data sample of 1.92 fb^{-1} accumulated at sqrt[s]=4.23 and 4.26 GeV with the BESIII experiment. When parametrizing the Z_{c}(3900)^{±} with a Flatté-like formula, we determine its pole mass M_{pole}=(3881.2±4.2_{stat}±52.7_{syst}) MeV/c^{2} and pole width Γ_{pole}=(51.8±4.6_{stat}±36.0_{syst}) MeV. We also measure cross sections for the process e^{+}e^{-}→Z_{c}(3900)^{+}π^{-}+c.c.→J/ψπ^{+}π^{-} and determine an upper limit at the 90% confidence level for the process e^{+}e^{-}→Z_{c}(4020)^{+}π^{-}+c.c.→J/ψπ^{+}π^{-}.