  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 GeV, 10.58 GeV, and 10.867 GeV,
respectively, we study the exclusive reactions $e^+e^-\to\gamma\chi_{cJ}$ ($J =
0,~1,~2$) and $e^+e^-\to\gamma\eta_c$. A significant $\gamma \chi_{c1}$ signal
is observed for the first time at $\sqrt{s}=10.58$ GeV with a significance of
$5.1\sigma$ including systematic uncertainties. No significant excesses for
$\gamma \chi_{c0}$, $\gamma \chi_{c2}$, and $\gamma \eta_c$ final states are
found, and we set 90\% credibility level upper limits on the Born cross
sections ($\sigma_{\rm B}$) 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 $\sigma_{\rm B}(e^+e^-\to\gamma\chi_{c1})$ is obtained.
