In the day's first game, Jeremy Guthrie scattered 10 hits and Stanford built an early lead before holding on for a 4-3 victory over Notre Dame.
Simmons (15-1) allowed one run and five hits in 7 1/3 innings, with his only costly mistake being a solo homer to Eric Arnold in the eighth.
Huston Street recorded the final five outs for his 11th save and was helped third baseman Omar Quintanilla.
Rice nearly tied the game in the eighth after two Texas errors. With runners on first and second and two outs, Quintanilla misplayed a ground ball by Paul Janish that went into foul territory. Vincent Sinisi rounded third and headed for home, but Quintanilla scooped up the ball and threw him out at the plate.
Simmons outdueled Steven Herce (13-3), who allowed two runs and eight hits in 7 2/3 innings.
Texas (54-15) advances to play Stanford in the winner's bracket Monday night. Rice (52-13) will play Notre Dame in the loser's bracket on Monday.
Texas opened the scoring in the third on singles by Quintanilla, Majewski and Jeff Ontiveros. The Longhorns made it 2-0 in the eighth when Majewski doubled home Tim Moss.
Guthrie improved to 13-1 with his win over Notre Dame. A junior righthander, he walked one and struck out five. He did give up a three-run homer to Javier Sanchez that cut Stanford's lead to 4-3.
"The numbers of our pitching staff are because of our defense," Guthrie said. "It gives us a lot of confidence. And getting the lead helps every time."
"The difference for us today was Jeremy Guthrie," Stanford coach Mark Marquess said. "He got a little tired but he came through for us. He came back and made good pitches when we needed it."
The Cardinal, which won back-to-back CWS titles in 1987-88, jumped to a 2-0 lead in the first on a two-run single by Jason Cooper. Chris Carter's RBI triple in the third made it 3-0.
In the sixth, Sam Fuld homered to make it 4-0 and chase Irish starter Grant Johnson, who allowed four runs and seven hits in 5 2/3 innings, walking two and striking out two. He tied a CWS record by hitting three batters.
"He showed a tremendous amount of poise to be 18 years old and in that kind of environment," Notre Dame coach Paul Mainieri said. "But against Guthrie you cannot give up that many runs."