  We observe for the first time the process $e^{+}e^{-} \rightarrow \eta h_c$
with data collected by the BESIII experiment. Significant signals are observed
at the center-of-mass energy $\sqrt{s}=4.226$ GeV, and the Born cross section
is measured to be $(9.5^{+2.2}_{-2.0} \pm 2.7)$ pb. Evidence for $\eta h_c$ is
observed at $\sqrt{s}=4.358$ GeV with a Born cross section of
$(10.0^{+3.1}_{-2.7} \pm 2.6)$ pb, and upper limits on the production cross
section at other center-of-mass energies between 4.085 and 4.600 GeV are
determined.
