This Letter reports the first scientific results from the observation of antineutrinos emitted by fission products of $^{235}\mathrm{U}$ at the High Flux Isotope Reactor. PROSPECT, the Precision Reactor Oscillation and Spectrum Experiment, consists of a segmented 4 ton $^{6}\mathrm{Li}$-doped liquid scintillator detector covering a baseline range of 7--9 m from the reactor and operating under less than 1 m water equivalent overburden. Data collected during 33 live days of reactor operation at a nominal power of 85 MW yield a detection of $25\text{ }461\ifmmode\pm\else\textpm\fi{}283\text{ }(\mathrm{stat})$ inverse beta decays. Observation of reactor antineutrinos can be achieved in PROSPECT at $5\ensuremath{\sigma}$ statistical significance within 2 h of on-surface reactor-on data taking. A reactor model independent analysis of the inverse beta decay prompt energy spectrum as a function of baseline constrains significant portions of the previously allowed sterile neutrino oscillation parameter space at 95% confidence level and disfavors the best fit of the reactor antineutrino anomaly at $2.2\ensuremath{\sigma}$ confidence level.