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.
5. Junior Firpo
5 - Found it difficult on the ball as well as off it. Gave it away too many times in the first half. Picked wrong options on occasion. Photo by Catherine Ivill/Getty Images.
6. Jamie Shackleton
6 - Busy in midfield, able to use pace to get Leeds upfield a few times. Not always accurate on the ball. Pretty solid at right-back. Picture by Bruce Rollinson.
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.