Where Andrew Luck and Jim Harbaugh failed, Kevin Hogan and David Shaw delivered.

On a wet night at Stanford Stadium, Stanford quarterback Hogan did just enough on offense and Shaw put his faith in kicker Jordan Williamson, who came through with a game-winning 36-yard field goal with five minutes left in the fourth quarter. Williamson’s second game-winning kick in three weeks broke a 24-24 tie with No. 17 UCLA (9-4) and sent No. 8 Stanford (11-2) to the Rose Bowl as Pac-12 champions, something former Stanford head coach Harbaugh and All-American quarterback Luck couldn’t do.

As the rain came down at a half-full Stanford Stadium, the Bruins jumped on the Cardinal early in a rematch of last Saturday’s regular-season contest, a Stanford win that was rather one-sided by the end. On Friday, however, running back Johnathan Franklin gashed the vaunted Cardinal defense for several big runs, including a 51-yard touchdown on the opening drive to serve notice that this week would be no cakewalk for Stanford.

The Cardinal responded quickly, however, with Stepfan Taylor doing some nice running of his own and Hogan fooling three-fourths of the stadium on a naked bootleg that let him stroll into the end zone to even the score at seven. It looked like the game might be setting up for a shootout, as Bruins quarterback Brett Hundley moved UCLA right back down the field and into the end zone put the visitors up 14-7 with 3:40 to play in the first quarter.

And UCLA seemed the aggressor, shoving the ball down Stanford’s throat on the ground with Franklin and Hundley and winning at the point of attack against the Cardinal’s offensive line. The Stanford defense seemed taken aback as well, not surprisingly as the Card had allowed just 20 points in the first quarter through the first 12 games of the season.

When the Bruins stopped Stanford and began driving into Cardinal territory, it seemed as though UCLA was firmly in the driver’s seat and that coach Jim Mora was pushing all the right buttons on his way to an upset. But an interception by junior safety Ed Reynolds, his conference-leading sixth of the year, was an absolute game-changer.

Reynolds came across the field and picked off Hundley’s underthrown pass around the 20-yard line, coming up inches short of a touchdown after a spectacular return.

“When you get a game-changing play like that, you have a good chance to win,” Shaw said, “because if the rest of the game is played even, that one is the one that tips it in your favor.”

A quick one-yard touchdown from Taylor knotted things at 14, and stemmed the tide that threatened to knock Stanford out of a BCS bowl.

The game settled down, however, as the defenses held their ground and it became quite clear that neither side was willing to take many deep shots in the air. Hogan finished 16-for-22 for 155 yards, but his leading receiver was Taylor on a multitude of check downs and few passes traveled more than 10 yards down the field. UCLA limited the effectiveness of tight ends Levine Toilolo and Zach Ertz, who made just four combined catches for 25 yards.

Stanford went into the break with the lead thanks to Williamson, who looked calm and confident in drilling a 37-yard field goal as time expired in the half.

“He’s been money,” Shaw said of his kicker. “He’s been really good, he’s got his routine down, he’s confident, he’s consistent, and that’s all I ever say to him. I just need him to be smooth and consistent, and that’s where he is right now.”

The second-quarter kick was preceded by a 10-yard run from Taylor that set the Stanford career record for rushing yards, moving the Mansfield, Texas, native past Darrin Nelson’s mark of 4,175, which had stood since 1981.

The Cardinal needed every one of Taylor’s yards, with Franklin putting up 203 on just 19 carries and Hundley adding 98 of his own. Shaw was one of those very impressed with the UCLA performance on the road.

“We were going to get best shot, and they gave it to us,” he said. “Their quarterback played great. Boy, he’s hard to tackle. Their running back made some big runs, their running back made big plays and they made big stops on defense, but our guys hung in there.”

Stanford received the second-half kick but could do nothing with it. The offensive line continued to do well in pass protection but failed to come up with any serious push in the ground game. The Cardinal gained just 170 yards on the ground on the evening, but that production came on 43 attempts and it was anything but easy running for Taylor throughout the night.

A Ka’imi Fairbairn field goal pulled UCLA even at 17, and the Bruins continued to run the ball well with Franklin breaking free for a 20-yard touchdown to cap a 12-play, 80-yard drive just before the end of the third quarter.

The Stanford students, who absolutely jam-packed the Red Zone, held their heads in their hands as the fourth quarter approached with the Cardinal down a score.

But Hogan found a way, driving the Cardinal back down the field for an immediate and impressive answer. Facing a third-and-15 on the Bruins’ 26-yard line, Hogan lofted a ball just beyond a UCLA cornerback’s fingertips and into the waiting arms of receiver Drew Terrell for a game-tying touchdown.

“There were four verticals … and Drew ran into that end zone, and Kevin held onto the ball to hold the safety inside as long as he could,” Shaw said. “He threw the ball, took a big hit, and it was a great throw, and a huge catch and a big play for us.”

The always-even-keeled Hogan said that he saw Terrell sneak past the corner and get into the back of the end zone on the play, before the quarterback found himself looking at the sky from his backside.

But that heroic strike merely tied the score with plenty of game remaining.

A couple of stalled drives on either side led to a UCLA punt that Terrell returned 18 yards into Bruin territory with a little over eight minutes to go in the game. Shaw then called Kelsey Young’s number on a sweep out of the wildcat formation, which went for 23 yards and put the Cardinal firmly in field goal position. From there it was all on Williamson’s right leg, and his kick from 36 yards easily sailed through and sent the crowd into a frenzy.

Despite an announced attendance of just over 31,000 fans, the raucous students that filled nine sections on their own had the near side of the field sounding just about as loud as any stadium in the conference usually gets.

UCLA couldn’t convert on third down on its ensuing drive, handing the ball back to Stanford with a few minutes remaining. But the Cardinal couldn’t run the clock out when Levine Toilolo was unable to handle a third-down pass, giving the Bruins one last chance.

A fourth-down pass to tight end Joseph Fauria kept UCLA’s drive alive with under a minute to play and the fans on their feet, but a potentially game-tying 52-yard field goal attempt by the Fairbairn went wide left with 39 seconds left and the stadium was smelling roses.

It came down to “character,” said Shaw, the Pac-12 coach of the year in his first two seasons. “Even when we don’t play well, we still play hard. Our guys played with such heart. We made plays when we needed to make plays.”

Stanford will take on the winner of the Big-10 Championship Game, which will take place tomorrow night between Nebraska and Wisconsin, in the Cardinal’s first Rose Bowl since the 1999 season, to be held in Pasadena on Jan. 1.