Jenkins became the first Jacksonville State player to pass for at least 300 yards since Nov. 3, 2012.
Barge finished with nine catches for a career-high 158 yards while Markis Merrill added 101 yards receiving on five receptions.
On its first possession, Jacksonville State (7-1, 5-0 Ohio Valley Conference) used an 11-play, 96-yard drive capped by a 14-yard touchdown pass from Jenkins to Barge. A 48-yard pick-6 by Dawson Wells, a 1-yard scoring run by Torymaine Pope and a 41-yard field goal by Connor Rouleau made it 24-0 at halftime.
Eastern Kentucky (5-3, 4-1), which came into the game having allowed five sacks this season, had just 175 total yards and gave up four sacks.
Jacksonville State's Miles Jones had 110 yards rushing.