The Bruins (33-3) came back from 10 points down in the second half to advance to the Sweet 16, where they will face either Western Kentucky or San Diego next week in Phoenix.
The game was tied 49-49 with 9.5 seconds left when Darren Collison made a layup in heavy traffic, and the UCLA defense denied the Aggies (25-11) attempt to tie it, when Josh Shipp stopped Donald Sloan's drive to the basket and forced a turnover in the closing seconds.
Russell Westbrook picked up the loose ball and drove the length of the court for a slam dunk as time expired.
Sloan finished with 12 points for the Aggies. Josh Carter scored 10.