Thompson's big day included 147 yards rushing and 11-of-20 passing for 127 yards.
The defense picked off 2016 Heisman Trophy victor Lamar Jackson four times.
Louisville beat Virginia and Syracuse to finish its ACC slate and then manhandled in-state rival Kentucky 44-17 in Lexington to close out the regular season.
He had the longest run of his career, a 75-yarder late in the second quarter, and a 13-yard TD scamper earlier in the game in which he split two defenders and made another look silly.
The Bulldogs marched right back down the field 79 yards to push ahead 14-7 on a 14-yard run by Thompson.
Jackson ran 24 times for 158 yards and a touchdown, breaking the TaxSlayer Bowl record for most rushing yards by a quarterback.
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Jackson, a Pompano Beach, Florida, native, has thrown for more than 3,400 yards and 25 touchdowns this season. But his desperation heave was batted away in the end zone.
The holiday season bowls just keep on coming, don't they?
Louisville enters this game with the exact same record, overall and conference, as the Bulldogs but the Cardinals have won three in a row.
Mississippi State enters play at 8-4 (4-4) on the season. The Bulldogs punched it in on the next play, taking a 31-27 lead after the extra point. Jackson even had a chance to win the game in the final minute, but Mississippi State's front seven, which had dominated most of the game, ended up winning the game on the field.
The Bulldogs weren't able to run out the clock and Louisville got the ball back with 1:39 left. The A-Train averages 4.6 yards per carry and has rushed for five scores this season. He has now done it twice (2016 and 2017).
He joined former Florida and Nevada quarterbacks Tim Tebow and Colin Kaepernick as the only others to accomplish that feat. He also has a rushing score, so that playing experience could be beneficial, especially considering the Cardinals open next season against perennial power Alabama in nearby Orlando on September 1.