Jennings hit Travin Dural with a 49-yard touchdown pass at the end of the length-of-the-field march to put the Tigers in front with 1:15 to play.
The young quarterback was called on in the fourth quarter after starter Zach Mettenberger suffered a knee injury. Earlier in the game Mettenberger had become the third quarterback in LSU history to throw for 3,000 yards in a career.
Arkansas came into the game as a four-touchdown underdog, but fought its way into the lead thanks to two scoring passes by Brandon Allen and two TD runs by Hunter Henry.
The Tigers closed the deficit to 27-24 when Zach Hocker kicked a 28-yard field goal with 9:27 to play and LSU soon had the ball again near midfield. But the Tigers failed to convert a fourth-down try, giving the ball back to the Hogs.
Arkansas, however, was forced to punt and that punt came to rest on the LSU 1-yard line. Jennings came in to save the game and give LSU a 9-3 regular-season record. The Tigers finished with a 5-3 mark in the Southeastern Conference.
Arkansas wound up at 3-9 and failed to win a conference game in eight tries.
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