Noyola, a sophomore midfielder who entered the match as a substitute midway through the first half, took the kick after Kelley O'Hara drew a foul in front of the penalty area. Noyola measured her shot carefully before chipping it off the fingertips of Santa Clara goalkeeper Bianca Henninger, who deflected it off the crossbar on a diving attempt, but the ball bounced past the goal line.
"If it had been an inch lower, she might have gotten to it," said Noyola, who scored her sixth goal of the season. "She did read it well, which made it even more exciting. I've known Bianca for a long time, played with her, played against her, and to get one on her was great. She's a great keeper."
In the first-round match against Northern Arizona, Noyola took a free kick from nearly the same spot and hit the post, giving her confidence she could score.
"It was a hell of a shot," Santa Clara coach Jerry Smith said. "For Bianca even to get there I thought was truly amazing. As I told our team today, what knocked us out of the tournament was the undisputed number one team in the tournament, and a hell of a free-kick shot that I think would've gone in against any goalkeeper."
The Cardinal had a prime opportunity to go up 2-0 when Noyola slipped a perfect centering pass to Christen Press. But the normally reliable junior missed the sitter, sending it high over an open net.
Another second-half chance - on an O'Hara header - was cleared off the line by Santa Clara's Kiki Bosio.
Instead, Santa Clara (14-7-2) applied much second-half pressure, with Stanford goalkeeper Kira Maker stretching to punch away the Broncos' best chance on a well-placed chip shot by Jordan Angeli in the 87th minute.
Santa Clara controlled much of the play during the latter part of the match.
"It was the longest 10, 12, minutes of my life," Noyola said. "I wasn't happy with how we ended the game," Stanford coach Paul Ratcliffe said. "We should've cleared the ball a little better, we didn't clear."
Stanford advanced to the quarterfinals for the seventh time in school history, and beat Santa Clara in NCAA play for the second time in seven meetings and first time since 1991. The teams met on Oct. 1, when Stanford beat the Broncos, 6-2.
"I don't think we thought we weren't the better team, because we knew we were the better team the whole time," Noyola said. "The playoffs are like that. It's sometimes a battle to the end. It's do or die for them, obviously, and us. That's how it's going to be for every team. We're not going to win 6-2 in the playoffs."