Louisiana Tech's Landry Lyddy threw an incomplete pass on fourth-and-10 to end it.
FIU (4-4, 2-2 Conference USA), which beat Charlotte 34-15 last week, has won consecutive games for the first time since 2019 and ended a four-game losing streak to Louisiana Tech (2-6, 1-3).
James was 31-of-48 passing for 321 yards and connected with three separate receivers for scores. Joseph finished with 41 yards rushing and also had a 1-yard TD run in the first overtime.
Chase Gabriel's 28-yard field goal pulled FIU even at 27 with 16 seconds remaining in regulation.
Lyddy threw for 256 yards and was 28-of-40 passing for Louisiana Tech. Marquis Crosby had 115 yards rushing and a pair of touchdown runs.