Rios was nearly thrown out at second for the third out of the ninth after his long fly to left hit off the wall. Texas almost got out of the inning again when Bowser hit a high fly, but it dropped about 15 feet from outfielder Dylan Campbell.
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Stanford (44-18) will play top-seeded Wake Forest on Saturday in the College World Series.
Bowser made it 2-0 in the second with his third home run in three games.
Stanford scored three runs with two outs in the fourth for a 6-3 lead. Carter Graham followed an intentional walk with an RBI single to break a tie and Braden Montgomery added a two-run single for a three-run lead.
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Texas (42-22) was also seeking its third consecutive appearance in the College World Series.
Campbell tied it at 6-all in the eighth with a single and then turned a double play on a long throw from right field to retire Saborn Campbell at third.