  The state $Y(2175)$ is observed in the process $e^+ e^- \to \eta Y(2175)$ at
center-of-mass energies between 3.7 and 4.6$\sim$GeV with a statistical
significance larger than $10\sigma$ using data collected with the BESIII
detector operating at the BEPCII storage ring. This is the first observation of
the $Y(2175)$ in this process. The mass and width of the $Y(2175)$ are
determined to be ($2135\pm 8\pm 9$)~MeV/$c^2$ and ($104\pm 24\pm 12$)~MeV,
respectively, and the production cross section of $e^+ e^- \to \eta Y(2175)\to
\eta\phi f_{0}(980)\to \eta\phi \pi^+ \pi^-$ is at a several hundred femtobarn
level. No significant signal for the process $e^+ e^- \to \eta' Y(2175)$ is
observed and the upper limit on $\sigma(e^+ e^- \to \eta' Y(2175))/\sigma(e^+
e^- \to \eta Y(2175))$ is estimated to be 0.43 at the 90\% confidence level. We
also search for $\psi(3686) \to \eta Y(2175)$. No significant signal is
observed, indicating a strong suppression relative to the corresponding
$J/\psi$ decay, in violation of the "12\% rule."
