  Using a data sample of $(10087\pm44)\times 10^6$ $J/\psi$ events collected by
the BESIII detector in 2009, 2012, 2018 and 2019, the electromagnetic Dalitz
process $J/\psi \to e^+ e^- \eta(1405)$ is observed via the decay $\eta(1405)
\to \pi^0 f_0(980)$, $f_0(980) \to \pi^+ \pi^-$, with a significance of about
$9.6\sigma$. The branching fraction of this decay is measured to be ${\mathcal
B}(J/\psi \to e^+ e^- \pi^0 \eta(1405) \to e^+ e^- \pi^0 f_0(980) \to e^+ e^-
\pi^0 \pi^+ \pi^-)=(2.02\pm0.24(\rm{stat.})\pm0.09(\rm{syst.}))\times 10^{-7}$.
The branching-fraction ratio ${\mathcal B}(J/\psi \to e^+ e^-
\eta(1405))$/${\mathcal B}(J/\psi \to \gamma \eta(1405))$ is determined to be
$(1.35\pm0.19(\rm{stat.})\pm0.06(\rm{syst.}))\times10^{-2}$. Furthermore, an
$e^+e^-$ invariant-mass dependent transition form factor of $J/\psi \to e^+
e^-\eta(1405)$ is presented for the first time. The obtained result provides
input for different theoretical models, and is valuable for the improved
understanding the intrinsic structure of the $\eta(1405)$ meson.
