We report the discovery of an eccentric hot Neptune and a non-transiting outer planet around TOI-1272. We identified the eccentricity of the inner planet, with an orbital period of 3.3days and Rp,b=4.1{+/-}0.2R{Earth}, based on a mismatch between the observed transit duration and the expected duration for a circular orbit. Using ground-based radial velocity (RV) measurements from the HIRES instrument at the Keck Observatory, we measured the mass of TOI-1272b to be Mp,b=25{+/-}2M{Earth}. We also confirmed a high eccentricity of eb=0.34{+/-}0.06, placing TOI-1272b among the most eccentric well-characterized sub-Jovians. We used these RV measurements to also identify a non-transiting outer companion on an 8.7day orbit with a similar mass of Mp,csin(i)=27{+/-}3M{Earth} and ec<~0.35. Dynamically stable planet-planet interactions have likely allowed TOI-1272b to avoid tidal eccentricity decay despite the short circularization timescale expected for a close-in eccentric Neptune. TOI-1272b also maintains an envelope mass fraction of fenv~11% despite its high equilibrium temperature, implying that it may currently be undergoing photoevaporation. This planet joins a small population of short-period Neptune-like planets within the "Hot Neptune Desert" with a poorly understood formation pathway.