Daniel Farke’s side were moments away from victory at the London Stadium after Ao Tanaka’s first-half goal and Dominic Calvert-Lewin’s second-half penalty put them in control.
But West Ham staged a stoppage-time rally as Mateus Fernandes reduced the deficit before Axel Disasi’s equaliser forced extra time.
With the score still level at 2-2 after the additional period, Leeds keeper Lucas Perri emerged as their shoot-out hero.
Joel Piroe missed Leeds’ first penalty but Calvert-Lewin, Brenden Aaronson, Wilfried Gnonto and Pascal Struijk all converted their kicks.
Perri saved Jarrod Bowen’s opening effort for West Ham and also denied Pablo as Leeds won 4-2 on penalties.
It was their second shoot-out victory in the competition this season following their fourth-round success at Birmingham.
Leeds will face Chelsea in the semi-finals at Wembley later in April.
“We always do it the tough way, never the easy way. We could have made our lives easier,” Farke said.
“We had a late sucker punch when the whole stadium was buzzing. To keep the nerves and win the penalty shoot-out shows great mentality and character.
“A chapter of Leeds history. It will be a big night for us at Wembley. It’s a great step for this club and a great reward for everyone.”
The Yorkshire club are in the semi-finals for the first time since 1987, when they lost to eventual winners Coventry at Hillsborough.
Southampton will face Manchester City in the other semi-finals after their shock win against Premier League leaders Arsenal.
The Championship side beat the Gunners 2-1 in the quarter-finals at St Mary’s on Saturday.
City thrashed Liverpool 4-0 on Saturday to reach the semi-finals of the competition for an eighth successive season.
Pep Guardiola’s side are bidding to make the final for a fourth consecutive year, having lost to Manchester United in 2024 and Crystal Palace in 2025 following their last win in 2023.
Southampton are dreaming of emulating their shock 1976 FA Cup final victory against Manchester United.
Relegated from the Premier League last season, Southampton are also hoping to win promotion via the Championship play-offs. (AFP)
Edited by Cecil Wong
