We report the Fermi LAT γ-ray detection of the 2021 outburst of the symbiotic recurrent nova RS Ophiuchi. In this system, unlike classical novae from cataclysmic binaries, the ejecta from the white dwarf form shocks when interacting with the dense circumstellar wind environment of the red giant companion. We find the LAT spectra from 50 MeV to ~20-23 GeV, the highest-energy photons detected in some subintervals, are consistent with π ⁰-decay emission from shocks in the ejecta as proposed by Tatischeff & Hernanz for its previous 2006 outburst. The LAT light curve displayed a fast rise to its peak >0.1 GeV flux of ≃6 × 10^-6 ph cm^-2 s^-1 beginning on day 0.745 after its optically constrained eruption epoch of 2021 August 8.50. The peak lasted for ~1 day and exhibited a power-law decline up to the final LAT detection on day 45. We analyze the data on shorter timescales at early times and found evidence of an approximate doubling of emission over ~200 minutes at day 2.2, possibly indicating a localized shock-acceleration event. Comparing the data collected by the American Association of Variable Star Observers, we measured a constant ratio of ~ 2.8 × 10^-3 between the γ-ray and optical luminosities except for a ~5×smaller ratio within the first day of the eruption likely indicating attenuation of γ rays by ejecta material and lower high-energy proton fluxes at the earliest stages of the shock development. The hard X-ray emission due to bremsstrahlung from shock-heated gas traced by the Swift-XRT 2-10 keV light curve peaked at day ~6, later than at GeV and optical energies. Using X-ray derived temperatures to constrain the velocity profile, we find the hadronic model reproduces the observed >0.1 GeV light curve.