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.
1. Illan Meslier
8 - Made a couple of terrific stops to keep it goalless. Not all of his distribution was entirely spot on. Picture by Bruce Rollinson.
2. Stuart Dallas
6 - Difficult start, took a few knocks and looked frustrated. Grew into the game. Drove forward from midfield in the second half. Photo by Catherine Ivill/Getty Images.
3. Kalvin Phillips
8 - Comfortable for the most part. Made important defensive contributions. Kept things simple in possession but tried to orchestrate from deep. Photo by George Wood/Getty Images.
4. Charlie Cresswell
8 - Saw a lot of the ball, did well with it in the main. Won headers, made blocks, performed solidly. His confidence grew as the game went on. Photo by Lewis Storey/Getty Images.
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.