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.
7. Mateusz Klich
7 - Some clever play in possession, linking attacks. Lots of work in defence against a mobile midfield. Photo by Catherine Ivill/Getty Images.
8. Dan James
7 - Looking more comfortable in the Leeds side. Posed a threat with the ball at his feet, helped to create. Worked back well too. Picture by Bruce Rollinson.
9. Crysencio Summerville
7 - Lively, wanted the ball, tried to get at Fulham. A couple of nice interceptions to put Leeds on the front foot. Photo by Catherine Ivill/Getty Images.
10. Rodrigo
7 - Missed a big chance in the first half but was heavily involved in a lot of Leeds' best attacks. Kept hunting the ball. Put in a big shift. Picture by Bruce Rollinson.
11. Tyler Roberts
5 - Lacked impact. Played up top in the first half and was unable to take the chance that came his way. Played as a 10 in the second half but was replaced after 10 minutes having contributed little. Picture by Bruce Rollinson.
12. Adam Forshaw (sub)
6 - Tried to get Leeds ticking and playing forward. A few inaccurate passes aside, did well. Photo by Catherine Ivill/Getty Images.