With Whites head coach Garry Monk having used all three substitutes, Lewie Coyle went down injured in the 90th-minute with the scores level at 1-1.
Alex Pritchard’s 14th-minute opener for Norwich had been cancelled out by Marcus Antonsson strike two minutes before the break and United then dominated the second half.
But with Leeds down to 10 men, Norwich steamed forward throughout extra-time and looked to have sealed progression when Nelson Oliveira clinically headed past Marco Silvestri in the 99th minute.
But depleted Leeds showed superb spirit to rally in the second period of added-time and made it 2-2 in the 110th minute when Chris Wood fired home from close range to take the game to a penalty shoot-out.
Leeds then held their nerve to book their place in the last eight with Marco Silvestri saving three spot-kicks before Ronaldo Vieira converted the winning kick to put United through.
In a frantic end-to-end game, Norwich had their first chance after just 12 seconds with Silvestri forced into a low save to deny Steven Naismith. Just five minutes later, Naismith fired over another decent chance.
Leeds won their first corner after 13 minutes yet the Whites set-piece actually proved the cue for Norwich to take the lead.
The Canaries cleared the delivery to Robbie Brady, who raced away down the left wing and delivered a dangerous cross that the onrushing Pritchard clinically finished with a diving header.
Leeds needed to up their game and did so with Alex Mowatt and Souleymane Doukara having half chances, but Norwich were only denied a second goal by fine goalkeeping from Silvestri who denied Josh Murphy, Brady and Oliveira in quick succession before Louis Thompson flashed an effort across goal.
Leeds were finally given some respite when winning a 39th-minute corner from which they looked to have equalised, but Kyle Bartley was adjudged to have handled the ball and was booked.
Norwich very nearly made it 2-0 when Silvestri tipped a low effort from Pritchard just wide.
But there was no denying Leeds two minutes before the interval when Kemar Roofe got the better of Ben Godfrey at the left-hand post and crossed for Doukara, who played the ball across goal for Antonsson to tap home.
Pritchard’s wicked delivery went just wide for Norwich before Leeds almost made it 2-1 when Antonsson’s rising shot was tipped just over by John Ruddy for a corner from which Bartley headed home for another disallowed effort, eliciting little protest.
Norwich went close at an early stage of the second period when Thompson ran through the Leeds midfield and fired wide.
Ruddy then produced a great save to keep out a Doukara header from a 55th-minute corner.
For Norwich, Oliveira had two chances to net but blazed one effort over before forcing a strong save from Silvestri.
Monk then played his ace card in sending on Chris Wood for Antonsson with 23 minutes left and his presence led to United creating the best chance of the match with Alex Mowatt presented with an open goal, but he hooked his effort wide. It proved Mowatt’s final act as Monk sent on Hadi Sacko.
Leeds continued to press with Wood’s low shot saved but Silvestri was tested by an 89th-minute effort from Sergi Canos, which the Italian held.
With extra-time looming Coyle went down injured and the right-back proved unable to continue, Leeds having used all three substitutes.
Norwich went for the jugular with Pritchard curling a smart effort just wide. A speculative low drive from Dorrans was defected for a corner and from it the Canaries regained the lead as Brady cleverly fed Pritchard outside the box who whipped in a fine cross that Oliveira headed home.
Amazingly Leeds came roaring back and made it 2-2 when Sacko countered down the right and delivered a fine low cross for Wood to poke home from close range.
The game went to penalties and Norwich’s Dorrans and United’s Wood converted their side’s first attempts. Pritchard then saw his effort saved by Silvestri before Roofe put Leeds 2-1 up. Naismith was also denied by Silvestri but Ruddy thwarted Phillips.
After Oliveira converted his attempt, Grimes blazed over the bar with the shoot out level at 2-2.
But Brady was also denied by Silvestri and 19-year-old Vieira tucked away his penalty in front of an ecstatic Kop.
Leeds United: Silvestri, Coyle, Bartley, Cooper, Berardi, Grimes, Mowatt (Sacko 76), Vieira, Roofe, Doukara (Phillips 81), Antonsson (Wood 67). Unused substitutes: Green, Taylor, Ayling, O’Kane.
Norwich City: Ruddy, Godfrey (Turner 77), Bennett, Bassong, Brady, Tetty (Dorrans 45), Thompson, Murphy (Canos 71), Naismith, Pritchard, Oliveira. Unused substitutes: Jones, Dorrans, Lafferty, Morris, Ramsay.
Referee: Andy Woolmer (Northants).