The Longhorns only reserve those for Omaha even though this program had every right to celebrate in that fashion after bouncing back from a 27-24 season in 2013, which saw them finish second-to-last in the Big 12.
Instead UT left fielder Ben Johnson secured the last out in his glove on a lazy pop fly and Texas' players celebrated briefly before filing in line to shake hands with each other and then their counterparts.
Getting to Omaha comes with the territory upon signing a national letter of intent at UT, and this Texas team has gotten them back for the 35th time in school history and the first time since 2011.
"This is why we came to school here," said starter Parker French, who threw six scoreless innings to beat Houston for the second time this season. "We came to play in big games like these. But our job is not done yet. We still have a lot of work to do. They say Omaha is its own tournament."
French, who was selected by the Detroit Tigers in the 19th round of the MLB Draft on Saturday, earned the win against Houston when the two teams faced each other on March 1, going seven innings while giving up one earned run on seven hits with seven strikeouts.
He was just as efficient on Saturday and received equal support behind him to help pitch the shutout. Texas relievers Travis Duke (1 2/3 innings pitched, three hits), Morgan Cooper (0.1 IP, 1 hit), and John Curtiss (1.0, 1 hit) finished off the game.
"Parker French was terrific as were the relief pitchers that came in and picked him up," Texas head coach Augie Garrido said. "I thought the pitchers pounded the zone and did their job. The guys are really playing for each other."
French struggled with his command at times early on despite giving up only two hits through the first four innings. But he got help from his defense when it mattered most.
That second hit came in the fourth in the form of a leadoff single from Michael Pyeatt, who moved to second on a sacrifice bunt. French followed that by walking Justin Montemayor before getting a momentous inning-ending double play turned by third baseman Zane Gurwitz, second baseman Brooks Marlow and first baseman Kacy Clemens.
The Longhorns immediately turned that into a game-changing four-run frame in the fourth that virtually signaled the beginning of the end of this Super Regional.
Ben Johnson started Texas' fourth with a double to left field that ate up Pyeatt, who had a good beat on the ball but couldn't quite get his glove up in time.
Payton, who reached base safely for the 101st straight game with a single to right field in the first inning, followed with a walk and then Tres Barerra, who isn't exactly fleet of foot, laid down a perfect infield bunt single to load the bases with no outs.
Houston got the force out at home on the next at-bat but couldn't contain C.J. Hinojosa, who singled in two runs on a two-strike pitch to right-center field.
"I was just trying to relax myself really," Hinojosa said. "I was looking for a pitch up, something I could hit in the air. I kind of expected something in."
Added Garrido: "It broke the game open and was the start of that momentum. They had the base loaded a couple of times and it didn't work out for them. The baseball gods weren't working out for them."
Collin Shaw followed with a sacrifice fly to right to make it 3-0 and Kacy Clemens singled in the fourth run for his first NCAA Tournament base hit.
French relied heavily on his defense the keep Houston from jumping right back in the game in the fifth when the Cougars worked the bases loaded with only one out.
After an error by Hinojosa and a hit batter by French, Clemens got a force out at home and Pyeatt grounded out harmlessly to Marlow to end the frame. Those last two outs came on back-to-back pitches.
"It just didn't happen," Houston head coach Todd Whitting said. "We ran out of gas. We just didn't get that hit. It's just that simple."
Houston couldn't get out of its own way on Saturday just as it couldn't at times on Friday. In its two games against UT this weekend, the Cougars left 18 runners on base, including 10 in scoring position.
"We just couldn't get that knock. It happens," Whitting said. "We could get that big hit to get us jumpstarted. The kids played hard until the last out."