Using data samples of 89.5 fb(-1), 711.0 fb(-1), and 121.4 fb(-1) collected with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider at center-of-mass energies 10.52, 10.58, and 10.867 GeV, respectively, we study the exclusive reactions e(+)e(-) -> gamma chi(cJ) (J = 0, 1, 2) and e(+)e(-) -> gamma chi(c). A significant gamma chi(c1) signal is observed for the first time at root s = 10.58 GeV with a significance of 5.1 sigma including systematic uncertainties. No significant excesses for gamma chi(c0), gamma chi(c2), and gamma chi(c) final states are found, and we set 90% credibility level upper limits on the Born cross sections (sigma(B)) at 10.52 GeV, 10.58 GeV, and 10.867 GeV. Together with cross sections measured by BESIII at lower center-of-mass energies, the energy dependency of sigma(B) (e(+)e(-) -> gamma chi(c1)) thorn is obtained.