Luck, who finished with 325 yards, now has thrown 53 TD passes in his career, moving past Jim Plunkett for third place on the school's all-time list. He trails only John Elway (77) and Steve Stenstrom (72).
Stepfan Taylor gained 153 yards on 22 carries for Stanford (3-0, 1-0 Pac-12), which has won 11 games in a row, the longest active winning streak in the nation after Auburn's 17-game run came to an end with Saturday's loss at Clemson.
Nick Foles completed his first 17 passes for Arizona (1-2, 0-1), but went 7- for-16 the rest of the way.
Luck appeared to be intercepted near the goal line on the second Stanford drive. The Cardinal caught a break when a video review overturned the call to an incomplete pass. Jordan Williamson came on to kick a 20-yard field goal to put Stanford on the board.
The Cardinal marched to the Arizona 24 on their next possession. On fourth- and-short, Anthony Wilkerson took a pitch and scampered down the left sideline for a touchdown.
After the teams traded field goals, Foles led the Wildcats on an 11-play, 79- yard drive that he finished with a six-yard strike to Juron Criner. Not only did the touchdown trim Arizona's deficit to 13-10 with 5:57 left in the second quarter, but it was the 46th of Foles' career, tying him with Tom Tunnicliffe for second all-time in school history.
Stanford moved into the red zone on the ensuing drive, but settled for a 33- yard Williamson field goal.
Arizona still trailed by six after Jaime Salazar missed from 45 yards as the first half ended. The woes continued for Salazar, who failed to convert a 36- yard attempt on Arizona's initial drive of the third quarter.
Later in the frame, Luck found Zach Ertz for a 16-yard TD.
Stanford was forced to punt on its next series, but a penalty prolonged the drive. And on the first play of the fourth quarter, Luck hooked up with Levine Toilolo for a 34-yard TD.
Jeremy Stewart capped the scoring and a 12-play drive that originated from Stanford's own nine-yard line with a two-yard TD run.
Toilolo led all receivers with 102 yards on four catches...Luck completed 20- of-31 throws...Foles had 239 passing yards...Stanford is 8-3 against the Wildcats since 1999.