We report the first electron neutrino cross section measurements on argon, based on data collected by the ArgoNeuT experiment running in the GeV-scale NuMI beamline at Fermilab. A flux-averaged $\nu_e + \overline{\nu}_e$ total and a lepton angle differential cross section are extracted using 13 $\nu_e$ and $\overline{\nu}_e$ events identified with fully-automated selection and reconstruction. We employ electromagnetic-induced shower characterization and analysis tools developed to identify $\nu_e/\overline{\nu}_e$-like events among complex interaction topologies present in ArgoNeuT data ($\langle E_{\bar{\nu}_e} \rangle = 4.3$ GeV and $\langle E_{\nu_e} \rangle = 10.5$ GeV). The techniques are widely applicable to searches for electron-flavor appearance at short- and long-baseline using liquid argon time projection chamber technology. Notably, the data-driven studies of GeV-scale $\nu_e/\overline{\nu}_e$ interactions presented in this Letter probe an energy regime relevant for future DUNE oscillation physics.