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Leeds was 2-0 up going into the 93rd minute of regular time after goals from Ao Tanaka and Dominic Calvert-Lewin, but Mateus Fernandes and Axel Disasi struck in a remarkable finale, leaving thousands of West Ham fans attempting to get back into London Stadium after leaving early.

West Ham had the better chances in extra-time but saw two goals disallowed for offside, while it also had to hand 20-year-old goalkeeper Finlay Herrick his debut off the bench after Alphonse Areola was injured late on.

Herrick saved the first kick he faced, but Jarrod Bowen and Pablo were denied by Lucas Perri as Leeds progressed from a tie it looked to have thrown away.

Both goalkeepers made fine saves in the first six minutes, with Areola tipping Noah Okafor's curler around the post before Perri brilliantly thwarted Valentin Castellanos.

But Leeds was exerting more control and deservedly took the lead in the 26th minute. Tanaka started and finished a sweeping move, turning Soungoutou Magassa before his shot deflected off Disasi and bounced in via the underside of the crossbar.

West Ham went agonisingly close to an equaliser when Castellanos headed Adama Traore's cross off the foot of the post, and Leeds went 2-0 up in the 75th minute as Calvert-Lewin kept his cool from the penalty spot, after Max Kilman's lunging foul on Brenden Aaronson.

Wilfried Gnonto sent a clever backheel trickling wide as Leeds hunted for a third, and it was made to pay for that miss in stunning fashion.

In the third of 11 added minutes, Bowen's ferocious effort bounced off the post and into the path of Fernandes, who blasted into the empty net. Three minutes later, there were wild celebrations among the home fans as Disasi stretched to turn Traore's cross home.

That remarkable fightback forced extra-time, where Castellanos and Pablo both had wonderful finishes disallowed for offside, the former after a terrible error from Perri and the latter after Bowen again rattled the woodwork from distance.

Leeds also went close as a combination of Areola and Tomas Soucek somehow denied James Justin on the line, and the West Ham goalkeeper would not make it to penalties as an ankle injury forced him to be replaced by youngster Herrick.

Herrick denied Joel Piroe from the opening kick, but Leeds was perfect from that moment on and Perri saved weak attempts from Bowen and Pablo as the visitors advanced.

When asked if his side could go all the way in the FA Cup, Leeds boss Daniel Farke wasn't about to get carried away.

"I said [in the past] it's not realistic," he admitted.

"We want to achieve unrealistic goals, and we are on the right path to do this.

"Step by step we want to celebrate a few more points in the Premier League.

"If we have a chance to play a second time at Wembley in the FA Cup this season we want to grab the chance with both hands."

Leeds's reward is a tie with eight-time winner Chelsea on the weekend of 25and 26 April after Monday's (AEST) draw, a clash Farke acknowledged would be difficult to win.

"We will be the underdog, that's for sure. It's a big name and a side full of top players. For that we are the underdog, but we have written history," he added.

"Nobody expected us to go through to the semi-final. And, of course, we are greedy to go to the next step.

"My focus now is on the Premier League. We are on such a good path and want to make sure we can celebrate there."

Leeds's attention will now turn swiftly to its Premier League survival bid, with a trip to rivals Manchester United up next on 14 April (AEST). 

"In terms of points, it doesn't help at all but each win is good for confidence, to know we can deliver under pressure and deal with setbacks," Farke said.

"There was criticism about scoring goals in the last games, so it was good we put this to bed. It's good for confidence and belief and spirit."

West Ham boss Nuno Espirito Santo could only urge his disappointed players to pick themselves up and focus on their own fight for Premier League survival.

"It will take more time to recover. The legs go when you win and recover quicker," Nuno said.

"We have the sadness and that will drag into this evening and tomorrow, but we have to react. We have important games, so we will keep fighting."

Nuno did, however, take the positives from the defeat to Leeds, but insisted that his team's never-say-die attitude could only take it so far.

"That and the way we prove to ourselves that we can never give up," he added.

"The game is not done until the referee says so, and we have to take that into our future starting on Friday [Saturday AEST]."