We report the discovery of TOI-172 b from the Transiting Exoplanet Survey Satellite (TESS) mission, a massive hot Jupiter transiting a slightly evolved G star with a 9.48-day orbital period. This is the first planet to be confirmed from analysis of only the TESS full frame images, because the host star was not chosen as a two-minute cadence target. From a global analysis of the TESS photometry and follow-up observations carried out by the TESS Follow-up Observing Program Working Group, TOI-172 (TIC 29857954) is a slightly evolved star with an effective temperature of T eff = 5645 ± 50 K, a mass of M ⋆ = {1.128}-0.061 0.065 M ⊙, radius of R ⋆ = {1.777}-0.044 0.047 R ⊙, a surface gravity of log g ⋆ = {3.993}-0.028 0.027, and an age of {7.4}-1.5 1.6 {Gyr}. Its planetary companion (TOI-172 b) has a radius of R P = {0.965}-0.029 0.032 R J, a mass of M P = {5.42}-0.20 0.22 M J, and is on an eccentric orbit (e={0.3806}-0.0090 0.0093). TOI-172 b is one of the few known massive giant planets on a highly eccentric short-period orbit. Future study of the atmosphere of this planet and its system architecture offer opportunities to understand the formation and evolution of similar systems.