Nottingham Forest also progressed after beating hosts Brighton on penalties in their quarterfinal. The game ended 0-0 after extra time and Forest won the shootout 4-3.
England international Eze grabbed the opening goal and created a second for Ismaila Sarr at Craven Cottage.
"That's what we expect from him, and that's why we love him," Palace defender Marc Guehi said.
"I tell him every day he's one of the best, but it's not about saying it, it's about showing it and he's done it."
Eddie Nketiah came off the bench and scored to complete the win for Palace, who can look forward to a semifinal at Wembley Stadium.
On Sunday, second-division Preston faces Aston Villa and Manchester City travels to Bournemouth.
With victory at Fulham, Palace recorded a club-record sixth away win in row.
"It's this group, the character of this group, the attitude of this group, the work rate of this group and this all gives us a belief that we can win every single game," Palace manager Oliver Glasner said.
Palace have never won the Cup and their best previous run ended in semifinal defeat to Chelsea in 2022.
The London club have reached the final twice — in 1990 and 2016 — losing on each occasion to Manchester United.
This year could be one of their best chances with so many of English soccer's leading clubs eliminated.
Eze's match-winning performance ended a memorable week for the forward, who also scored for England in Monday's World Cup qualifying win over Latvia.
Cutting in from the left, he curled a low shot from outside the box that went in off the post to fire Palace ahead in the 34th minute.
Four minutes later, he crossed for Sarr to head in a second for the visitors.
Substitute Nketiah fired through Fulham goalkeeper Bernd Leno's legs to seal the win in the 75th.
Another bonus for Palace was the return of Jean-Philippe Mateta, who played for the first time since a serious head injury required 25 stitches on a severely lacerated ear.
The forward was hospitalized after being kicked in the head by Millwall goalkeeper Liam Roberts in the previous round of the FA Cup this month.
Wearing a protective covering on his ear, the French forward played for 70 minutes before being replaced by Nketiah.
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