Berry broke the Lamar career record with his 41st touchdown pass, breaking John Evans' mark of 40. He now has 43 career TD passes.
Mark Roberts caught three passes for 139 yards and two touchdowns and Reggie Begelton pulled in 12 passes for 119 yards and one TD.
Berry put the Cardinals in front early when he hit Roberts on a 69-yard scoring strike with 11:32 remaining in the first quarter. Grambling came back to take a 10-7 lead following Dre' Fusilier 2-yard run and Marc Orozco's 41-yard field goal.
The Cardinals took control in the third quarter, scoring 28 points, to lead 42-13.
Grambling, playing under first-year coach Broderick Fobbs, scored twice more as Jestin Kelly ran in from the 2 and Stephen Johnson scored from the 4 with 10:17 remaining.