With West Ham sitting 18th in the Premier League, in the relegation zone and in danger of dropping to the Championship, this was a glorious chance for the players to give the supporters a day out at Wembley.
Before the game, captain Bowen said the club could turn "an underwhelming season into a really special one" and said the match was a "big opportunity".
So often the Hammers' star performer this season, Bowen added: "A Wembley trip for everyone associated with this club would be incredible. This is the FA Cup, the competition we all grew up watching and dreaming of winning."
But, on a desperately disappointing afternoon for home fans, the Hammers could not take their chance.
A poor first half featured them giving the ball away sloppily on several occasions, with discontent rising in the stands.
Bowen had a shot saved following a bright run by Adama Traore, but it was no surprise when Leeds went ahead. The tie then looked over when Calvert-Lewin added a second, with a penalty that was given following a check by the video assistant referee.
But, in a barely believable stoppage-time period, West Ham scored twice to earn a second bite at the cherry. Fernandes followed up after Bowen hit the post and then Disasi steered in Traore's cross.
In a breathtaking game that had nearly everything, West Ham then had two goals ruled out in extra time – Taty Castellanos and Pablo each offside when they found the net, the latter after Bowen rattled the woodwork with a fierce shot.
Then came the frustration of the penalties, and West Ham must quickly pick themselves up before Friday's home league game against bottom-placed Wolves, a fixture where Nuno Espirito Santo’s side will see victory as essential if they are to stay in the top flight.
