Stanford (25-0-0) plays North Carolina in Sunday's final at 1 p.m. ET.
Press was in the right position when a long ball played from the Stanford defense caromed off a UCLA defender, and the junior fired a low shot inside the far post for her 21st goal of the season.
"I remember thinking through the whole overtime that, as a forward, I would have a chance," Press said. "I had to be ready for my one opening. When it comes, I had to hit it hard and low."
Press has set up Kelley O'Hara for her 26th goal of the season -- tops in the country -- in the 64th minute to give Stanford the lead, but UCLA evened the score 12 minutes later when Lauren Cheney fired a shot that eluded Stanford goalkeeper Kira Maker.
In the final seconds of regulation, O'Hara almost won the game but she hit the right post.
"It was a letdown when we let the goal in, but we've been in this position before," O'Hara said. I thought we were playing well up until that point. I think we responded well to it, obviously. We went back at them and got a few more chances, and went into overtime feeling the momentum."
The loss marked the seventh year in a row UCLA reached the Women's College Cup but will return home without the national championship.