Leeds created a host of opportunities in a goalless first half, the best chance falling to Rodrigo who could not get a proper connection to a shot from close range after a Tyler Roberts effort was saved.
Whites head coach Marcelo Bielsa then handed a Leeds debut to teenage forward Joe Gelhardt from the bench after the break.
Another teenager, Stuart McKinstry, was then handed his Whites debut when replacing full debutante Crysencio Summerville but neither side could find a breakthrough.
It meant the game was decided by spot kicks and keeper Illan Meslier proved the hero by saving a penalty from Rodrigo Muniz as the shoot out went to sudden death.
Youngsters Gelhardt and McKinstry both kept their cool to brilliantly convert their efforts from 12 yards to help put Leeds into the last 16.
Graham Smyth, the YEP's chief football writer, hands out the scores.
13. Joe Gelhardt (sub)
6 - Raced around, chasing down the keeper and defenders. Tried to get on the ball but didn't really get the service to do any damage on his debut. What a penalty. Photo by Adam Davy - Pool/Getty Images.
14. Stuart McKinstry (sub)
6 - Wanted the ball, found space and caused a problem or two. Came inside to close Fulham down when necessary. A classy penalty. Photo by George Wood/Getty Images.
15. Tim Robinson (referee)
9 - An unfussy display. Didn't miss much for either side, refereed it sensibly. Had little controversy to deal with at all.