LaGow, who passed for 289 yards, connected with Ian Thomas for a 4-yard touchdown with 48 seconds left in the second quarter to increase Indiana's lead to 14-0 after Morgan Wilkerson broke a scoreless tie with a 2-yard run with 6:18 remaining in the first half.
Indiana (4-6, 1-6) keeps its chances of a third consecutive bowl game alive if it can beat Rutgers and Purdue in its final two games.
Illinois (2-8, 0-7) cut the deficit to 14-7 on the first play of the second half on George's 77-yard touchdown pass to Louis Dorsey.
A 28-yard Griffin Oakes field goal with 5:13 left in the third period gave the Hoosiers a 17-7 lead.
George countered with a 14-yard touchdown pass to Caleb Reams with 12:53 left in the game to pull the Illini within 17-14.
But LaGow's 5-yard touchdown pass to Simmie Cobbs with only 5:45 left essentially iced it for the Hoosiers.
Indiana had not beaten a Big Ten opponent since Purdue since the 2016 final.