Playing his fifth college game, Wall capped a big night with a slam dunk at the final horn, setting off a wild celebration by Kentucky players before a crowd of 1,425.
Wall hit a 12-foot fadeaway jumper from the right wing to tie the score at 61 with 30.6 seconds remaining in regulation, then two free throws with 2.4 seconds left to send the game to overtime.
Fellow freshman Eric Bledsoe made an NBA-length 3-pointer with 33 seconds to go that gave Kentucky a 69-65 lead it did not relinquish.
The Wildcats opted not to call a timeout and went to freshman forward DeMarcus Cousins, who was fouled on the block. The 53.6 percent free throw shooter missed the first free throw and then the second on purpose, but fouled Stanford's Jarrett Mann on the rebound.
Mann, a 53 percent free throw shooter, then missed both of his free-throw attempts, setting up a final flurry. Wall drew a blocking call on Andrew Zimmermann with 2.4 seconds left and calmly swished both free throws to tie it again.
Stanford's full-court pass was batted off the backboard by Patrick Patterson and corralled by Wall to send the game to overtime.
The game was marked by several long stoppages in play. Officials spent several minutes reviewing a Stanford 3-pointer and intentional foul in the first half, then a shot clock malfunction had tournament organizers searching for a ladder with 11:31 left in regulation.