The Eagles took the lead after a nervy start when Marc Guehi, who was called up to Gareth Southgate’s senior England side in midweek, was left unmarked inside the area and guided a Michael Olise corner past Jordan Pickford.
Jean-Philippe Mateta doubled Palace’s lead in the 41st minute when he turned home Wilfried Zaha’s cross with a low and powerful finish.
Zaha confirmed Palace’s place at Wembley in the 79th minute when Michael Olise’s shot rebounded off the post, and the Ivorian confidently tucked his shot home.
Will Hughes added a fourth with three minutes of normal time remaining – Conor Gallagher’s shot was saved by Pickford and fell to him with the goal at his mercy.
The Eagles have reached the semi-finals of the FA Cup for the first time since the 2015-16 season under Alan Pardew.
Liverpool and Manchester City still have quarter-final ties to play, and Chelsea booked their place in the semi-final after seeing off Middlesbrough.
When asked if anything was possible now Palace are in the last four, manager Vieira said: “Anything is possible, yes. But there are still quarter-finals to play. Depending on who goes through, but looking at the draw, they will all want to play Palace.
“If Chelsea, City and Liverpool go through, they will want to play us because we will be the underdogs.
“When you get to that level, anything can happen, and that is the beauty of our game.”
With Jeffrey Schlupp suspended, Vieira picked an attacking starting line up and handed Eberechi Eze his first since the 1-1 draw with Brighton in mid-January.
“He didn’t play much because the players in the midfield have been doing really well,” said Vieira. “He did really well – he played with a lot of personality. I was really pleased with the work he did out of possession.
“That part of the game he needed to improve, and he did it really well. Now he needs to be consistent, and the best way to do that is you need to play games.”