Afterward, Stanford's players cheered wildly in their victorious locker room, even belting out the national anthem at the top of their lungs.
"We celebrate all victories, and this one's a big one," quarterback Kevin Hogan said.
Hogan threw four of his five scoring passes to Montgomery, including a 9-yard completion just before halftime that put Stanford ahead 42-13. The Cardinal (9-2, 7-2 Pac-12) bounced back from last week's loss at USC but needed No. 5 Oregon to lose one of its final two games to win the Pac-12 North. The Ducks lost 42-16 in Tucson, and Stanford's fans led chants of "Arizona! Arizona!"
Stanford beat Oregon, giving the Cardinal the tiebreaker.
Hogan set career highs with 329 yards passing and the five TD passes.
Cal (1-11, 0-9) lost its 10th straight to finish coach Sonny Dykes' disappointing debut year.
"I can make a bunch of excuses, but my job is to get these guys ready to play," Dykes said, "and obviously I haven't."
Michael Rector caught a 45-yard touchdown pass midway through the second quarter for Stanford, while Tyler Gaffney ran for a 58-yard score out of the wildcat formation in the third for a 49-13 lead — giving Stanford its season high for points and most ever in a Big Game. The Cardinal were 31?-point favorites, and the margin of victory was largest in Big Game history.
Other than that, it was Montgomery's day. He tied Darrin Nelson's single-game touchdowns record set at Oregon State in 1981.
He ran for a 31-yard touchdown and caught a 50-yard touchdown pass both in the opening 3 1/2 minutes, then added a 12-yard TD reception with 6:58 to go in the first quarter. On the second play of the second quarter, he made a 72-yard catch and run from Hogan, who was credited with a career-long completion on the play.
Montgomery scored touchdowns on his first three catches of the day, and he had five receptions for 160 yards by halftime.
He had his first two touchdowns on only three offensive plays run by the Cardinal to start the game.
"It's awesome, just getting him the ball and he makes me look good," Hogan said.
Hogan completed his first five passes and finished 17 of 26 in Stanford's 15th straight home victory and fourth in a row in this rivalry. Backup Evan Crower took over late in the third and threw a 14-yard touchdown pass to Francis Owusu.
Stanford, which hosts Notre Dame next weekend to conclude the regular season, finished with 603 total yards and 417 yards passing.
Cal quarterback Jared Goff left the game with 7:06 remaining in the first half with a separated right shoulder after a hard hit by linebacker Shayne Skov, who also clobbered him earlier. Goff landed awkwardly and stayed down for a few minutes, then walked off on his own accompanied by trainers and headed for the locker room.
"After the first one, I told them I wanted to play, so I went right back in," Goff said.
Zach Kline played the rest of the way in Goff's absence, going 8 of 14 for 115 yards.
Goff went 10 for 19 for 194 yards, giving the freshman 3,508 yards passing to break Pat Barnes' previous single-season record of 3,499 set in 1996.