The Transiting Exoplanet Survey Satellite TESS has begun a new age of exoplanet discoveries around bright host stars. We present the discovery of HD 1397b (TOI-120.01), a giant planet in an 11.54-day eccentric orbit around a bright (V = 7.9) G-type subgiant. We estimate both host star and planetary parameters consistently using EXOFASTv2 based on TESS time-series photometry of transits and radial velocity measurements with CORALIE and MINERVA-Australis. We also present high angular resolution imaging with NaCo to rule out any nearby eclipsing binaries. We find that HD 1397b is a Jovian planet, with a mass of 0.415 ± 0.020 M_J and a radius of 1.026 ± 0.026 R_J. Characterising giant planets in short-period eccentric orbits, such as HD 1397b, is important for understanding and testing theories for the formation and migration of giant planets as well as planet-star interactions.