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−→γχcJ (J=0, 1, 2) and e+e−→γηc. A significant γχc1 signal is observed for the first time at s=10.58 GeV with a significance of 5.1σ including systematic uncertainties. No significant excesses for γχc0, γχc2, and γηc final states are found, and we set 90% credibility level upper limits on the Born cross sections (σB) 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 σB(e+e−→γχc1) is obtained.