Ballage equaled the mark of Howard Griffith, who scored eight times for Illinois in a 1990 win over Southern Illinois.
Ballage, a junior from Peyton, Colorado, rushed for seven touchdowns on 13 attempts, gaining 137 yards. His 75-yard run (a career long) with 9:11 to play in the fourth quarter allowed him to match the record.
He also caught a 39-yard touchdown pass from Manny Wilkins in the second quarter and finished with 185 all-purpose yards.
"It wasn't me -- it was God and my offensive linemen," Ballage said in an on-field interview after the game.
He later brought his entire offensive line with him to the postgame news conference.
"These guys right here, every single one of them, they go to work," Ballage said.
Ballage had scored nine touchdowns in his career before Saturday, including one on the ground in the Sun Devils' season-opening win over Northern Arizona.