The previous record was 448, set by Carson Palmer against Oregon in his Heisman Trophy-winning season in 2002.
"It's definitely an honor, and it feels great," Barkley said afterward. "I grew up looking up to Carson Palmer and hoping one day I'd be in this position.
"It's definitely a dream."
Barkley, a junior, previously broke the school record for completions with 34 in the Trojans' season opener and tied the school record for scoring passes in a game with five in the third game of the season against Syracuse.
He opened Saturday's game against the Wildcats with 186 yards on nine pass attempts in the first quarter, including an 82-yard catch-and-run toss to receiver Robert Woods. By halftime, he had 301 yards and three touchdowns, including a 38-yard touchdown to freshman Marqise Lee.
Barkley threw four more times to Woods in the third to finish that quarter with 382, and then broke the record on a 25-yard pass to Brice Butler in the fourth.
"It didn't feel like I was breaking a record because it was such a tight game going back and forth," Barkley said.
Woods, a sophomore, was 5 yards away from tying the school record for receiving yards in a game with 255 yards on 14 catches.
The previous record of 260 was set by then-freshman R. Jay Soward on just six catches against UCLA in 1996.
Pedro Moura covers USC for ESPNLosAngeles.com.