The Ragin' Cajuns (2-3, 1-0 Sun Belt) secured the win when Tracy Walker made an open field tackle two yards short of a first down on a fourth-and-6 play with 1:07 left.
The game saw five lead changes in the second half.
Marcus Caffey's third touchdown, a 2-yard run with 6:35 to play, gave the Panthers (1-4, 0-2) a 31-28 lead with 6:35 to play.
Louisiana-Lafayette responded with a seven-play, 75-yard. Broadway finished 20 of 26 for 216 yards. Robinson had eight receptions for 119 yards.
Wil Lutz had a 48-yard field goal as time expired in the first half gave Georgia State a 17-14 lead.
Alonzo Harris and Caffey each had two touchdowns in the second half.