After 90 minutes produced a goalless stalemate between the two sides – punctuated by missed chances from both sides – Fulham blinked first in the spot kicks as Josh Onomah missed its first effort.
A further miss from Joe Bryan had home nerves jangling, but Stuart Dallas and Adam Forshaw also failed to convert for the visitors, and ultimately it went into sudden death with Leeds boss Marcelo Bielsa barely able to watch on the sidelines.
Two rounds of flawless kicks followed, but after Stuart Adam McKinstry gave Leeds the advantage at 6-5, Illan Meslier dived to his left to save Rodrigo Muniz attempt and Bielsa's could breathe again as his side took the tie.
A little further west in London, Brentford made light work of its League Two opponent Oldham Athletic, thrashing the visitors 7-0 at Brentford Community Stadium.
Marcus Forss scored four of the goals, including a third-minute penalty to start the rout. Yoane Wissa also got on the scoresheet twice, and Raphael Diarra put the ball into his own net just before half-time to make it a humbling night for the Latics.
Burnley cruised past local rival Rochdale 4-1 despite going behind to a 47th-minute strike from Jake Beesley. Jay Rodriguez was the star, scoring all four of his side's goals as the Clarets ultimately ran away with it at Turf Moor.
Preston North End also won by a 4-1 scoreline, knocking Cheltenham Town out with a flurry of late goals at Deepdale.
Andrew Hughes and Joe Rafferty put Preston ahead, but Kyle Vassell gave the visitors some hope with a 56th-minute strike. Sean Maguire made it 3-1 with eight minutes to go, though, before Emil Riis Jakobsen wrapped it up in stoppage-time.
Elsewhere, Southampton eventually got past Sheffield United, winning the night's third penalty shootout 4-2 after the sides drew 2-2 in normal time, while Stoke City upset Watford 3-1 at Vicarage Road to stamp its ticket into the next round.
Sunderland also succeeded away from home, beating Wigan Athletic 2-0 in their tie at DW Stadum.