  The detection of collider neutrinos will provide new insights about neutrino
production, propagation, and interactions at TeV energies, the highest
human-made energies ever observed. During Run 3 of the LHC, the FASER
experiment is expected to detect roughly $10^4$ collider neutrinos using its
emulsion-based neutrino detector FASER$\nu$. In this study, we show that, even
without processing the emulsion data, low-level input provided by the
electronic detector components of FASER and FASER$\nu$ will be able to
establish a $5\sigma$ discovery of collider neutrinos with as little as
$5~\text{fb}^{-1}$ of integrated luminosity. These results foreshadow the
possible early discovery of collider neutrinos in LHC Run 3.
