Virginia ace Danny Hultzen and Tyler Wilson were within an out of combining for the first CWS shutout in five years before Chad Bunting hit a run-scoring single over shortstop with two outs in the ninth inning.
Hicks' one-out base hit into center ended the longest scoreless CWS game in 24 years, and Steve Proscia followed with a sacrifice fly.
The No. 1 seed Cavaliers (55-10) move to a Bracket 2 winners' game on Tuesday night. They'll play the winner of Sunday night's Texas A&M-South Carolina game.
Cal (37-22) meets the loser Tuesday.
Hultzen, the No. 2 overall draft pick by the Seattle Mariners, allowed three hits over 6 1-3 innings. He walked three and struck out six. Wilson (9-0) got the win and Branden Kline recorded the last out for his 18th save.
Logan Scott (1-2) took the loss in relief of Erik Johnson, who struggled for a third straight start.
Virginia was 1 for 11 with runners in scoring position through six scoreless innings.
The Cavaliers finally broke through after No. 9 hitter Keith Werman singled leading off the seventh and Chris Taylor walked, chasing Scott and bringing on closer Matt Flemer.
Virginia added to the lead in the eighth on Jared King's RBI triple and Werman's run-scoring single.
Cal finished sixth in the Pac-10 and was a surprise CWS qualifier after the school administration threatened to drop the program next year. A $9 million fundraising effort saved the program.
Virginia is trying to win the ACC's first national title in baseball since Wake Forest 56 years ago.
Monday, June 20
Texas vs. NORTH CAROLINA, Omaha, Neb. College World Series, 2:00 PM
Tuesday, June 21
VIRGINIA vs. South Carolina, Omaha, Neb., 7:00 PM