Danny Hultzen and Tyler Wilson were within an out of combining for the first CWS shutout in five years before Chad Bunting dropped a run-scoring single over the shortstop with two out in the ninth.
Hicks' one-out hit into center ended the longest scoreless CWS game in 24 years and Steve Proscia followed with a sacrifice fly.
The Cavaliers (55-10) advanced to a Bracket 2 winners' game Tuesday night against South Carolina. Cal (37-22) will face Texas A&M Tuesday afternoon.
Hultzen, the No. 2 overall pick by the Seattle Mariners, scattered three hits over 6.1 innings. Wilson (9-0) got the win and Branden Kline earned his 18th save. Logan Scott (1-2) took the loss.