Fourth-seeded Oklahoma (11-7-2, 5-4-1) will play the winner of the 5:30 p.m., match between top-seeded Texas A&M and No. 8 Colorado. The semifinal is Friday at 5:30 p.m.
Mooney, a sophomore from Edmond, Okla., played in three NCAA Tournament games last year while enrolled at Maryland. The transfer has seven goals this season for the Sooners.
"Caitlin is a great finisher," Sooners coach Nicole Nelson said. "She's extremely composed in front of goal. She put it in a position that was tough for Alexa (Gaul, the Longhorns goalkeeper) to make a save."
Fifth-seeded UT (11-5-4, 4-3-3) had an edge in the stats, but still bowed out of the first round for the second time in three years. The Longhorns had seven shots on goal to OU's four, led in corner kicks 5-2 and led in shots 14-11.
"It's been a struggle for this team to score, and that bit us today." Texas coach Chris Petrucelli said. "Mooney did a pretty good job with her back to the goal and spun and hit a pretty good shot for them.
"Honestly, I don't know if they had very many other opportunities in the game. They had one chance and they put it away, and we had three or four and didn't."
The Sooners had defeated UT in their only previous meeting of the year, 2-1 in double overtime.
The game was played with a steady north wind at 25 mph, which the Longhorns were unable to capitalize upon when they had it at their back the first half. They did outshoot OU, but the Sooners led 1-0 at halftime on Mooney's goal in the 30th minute.
"I think (the wind) played to our advantage because we're more of a technical team and were able to move the ball around," Mooney said.
Mooney's goal was assisted by Whitney Palmer, who advanced the ball up the left side and found Mooney cutting about 20 yards right in front of the goal.
"(Palmer) brought four of their defenders with her," Mooney said. "I just wanted to go in support for her if she was going to cross it or play it back. She played me back and I was able to turn and shoot."
That was OU's only shot on goal of the opening half, and the best of UT's three shots to reach the goal was midfielder Kylie Doniak's low kick that Kelsey Devonshire was able to deflect out of bounds in the 16th minute. UT generated nothing from the ensuing corner kick, one of their two of the first half compared to none for the Sooners.
Devonshire, a sophomore from North Richland Hills, Texas, contributed to OU's fifth shutout win this year.
Doniak, a junior from Chino Hills, Calif., came into the tournament riding a six-match scoring streak, which had been the longest for UT since Niki Arlitt scored in six straight during the 2007 season.