But the Sun Devils had a rally of their own, scoring three touchdowns in the final 4:39 to remain firmly in the playoff mix.
Notre Dame is all but out of the playoff hunt after Golson threw four interceptions and lost a fumble.
Kelly threw for 224 yards on 17-of-28 passing and capped one of Arizona State's biggest wins as a program with a last-second, 2-yard scoring run.
Golson finished with 446 yards on 22-of-41 passing and two touchdowns, but has 17 turnovers the past six games, including two interceptions returned for touchdowns by the Sun Devils. He also was sacked seven times.
With all that was on the line, this was arguably the biggest regular-season game in the 56-year history of Sun Devil Stadium.
Students began camping out for tickets Monday and fans jammed the stadium, creating a playoff-like atmosphere in the desert.
And it was like a playoff for the playoffs: Win and bolster your resume for the final rankings on Dec. 7, lose and start angling for a lower-tier bowl.
Notre Dame won the first game in the series last season, pulling out a wild 37-34 victory in Texas in a game that featured two touchdowns in the final 68 seconds.
The Sun Devils took control of the rematch with two turnovers in 12 seconds.
Tashon Smallwood handled the first one on his own, racing around the left end to sack Golson -- ASU's third sack in four plays -- then dived on the ball the Notre Dame quarterback fumbled. Jaelen Strong scored the next play, spinning around to make a one-handed, 13-yard catch.
On Notre Dame's next play from scrimmage, Sun Devils defensive tackle Demetrius Cherry leaped to swat Golson's pass into the air and Marcus Hardison came up with it for an interception at Notre Dame's 23. Richard scored three plays later on a 1-yard run.
Hardison helped keep the onslaught going, deflecting a pass that fell into the arms of Damarious Randall, who raced in 59 yards for a touchdown that sent a roar through the stadium.
As statements go, a 24-3 lead by the opening seconds of the second quarter was a pretty big one.
Once the second half got going, so did the Irish.
Even after Golson threw an interception in the end zone on the opening drive, Notre Dame picked apart Arizona State's defense with one big play after another.
Cam McDaniel sandwiched a pair of 1-yard touchdown runs around a fumbled snap by holder Hunter Smith on a short field goal, and Amir Carlisle was left wide open for a 25-yard touchdown that pulled the Irish within 34-31.
But that was it for the Irish.
Kelly marched Arizona State down the field to set up his scoring pass to Richard, then Notre Dame receiver Corey Robinson bobbled a pass from Golson and the ball fell into Carrington's arms to set up the pick six.
Kelly sealed it by scampering around right end for his touchdown run.