Powell scored 12 points and had 15 rebounds in the win, which sent the Cardinal (19-14) on to a quarterfinal match-up with Alabama.
Aaron Bright added 11 points for Stanford, including three 3-pointers.
Taylor Smith led Austin (27-5) with 18 points and seven rebounds. Antonio Bostic scored 15 points and Hal Bateman added 12 plus seven assists.
Stanford led 58-52 with just less than 4 minutes to play, but the Lumberjacks stiffened on defense and cut the lead to one. Desmond Haymon took the final shot at the buzzer for the win but came up short.