Joe Gelhardt and Stuart McKinstry both hit the target as the penalty shootout reached sudden death after a goalless draw at Craven Cottage.
Gelhardt had missed from the spot just two days prior for the Under 23s at Liverpool, but ignored the noise from the Fulham fans at the Hammersmith End of the ground to hammer the ball high into Marek Rodak's net.
McKinstry took the next one for Leeds and placed it low into the far left hand corner, giving the keeper no chance.
Illan Meslier then saved from Rodrigo Muniz to give Leeds 6-5 shootout victory and a place in the fourth round.
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Bielsa felt it was an 'average performance' and Fulham made life difficult for his side to pass out from the back. The number of chances created by the Championship side, who boasted 11 changes to their starting line-up, concerned the head coach as well.
But he was delighted to see his side through to the next round, with help from two youngsters making their senior debuts.
"They showed personality and character," said Bielsa, who chose the penalty takers and arranged the order in which they would step up.
"Gelhardt had missed his last two penalties, he took on board the responsibility and managed to score. McKinstry also had to take one in the most decisive moment and they showed character."
Leeds were without the injured Robin Koch and Diego Llorente, with Pascal Struijk out injured, and both Patrick Bamford and Luke Ayling have joined the injured list.
Bielsa was unable to confirm if either Bamford or Ayling will return to face West Ham United on Saturday at Elland Road.
"Bamford has an injury in his ankle and Ayling in his knee," he said.
"It's subject to the evolution that they have."