Most of the second half was played in Palace's half as Fulham tried to respond but the hosts were caught cold again when substitute Eddie Nketiah fired a shot past Bernd Leno in the 75th.
While Oliver Glasner's Palace can look forward to a semifinal at Wembley and dream of winning the FA Cup for the first time in the their history, Fulham will console themselves with a push for Europe via the Premier League.
Also on Saturday, Nottingham Forest secured a place in the semifinals with a 4-3 penalty shoot-out win against Brighton & Hove Albion when Ryan Yates smashed home the winner after the game finished goalless after extra time.
Forest keeper Matz Sels saved Jack Hinshelwood's low shot towards the bottom corner before keeping out Diego Gomez's attempt into the middle of the goal either side of the visitors' missed effort from Neco Williams that flew over the bar.
On Sunday, second-tier Preston North End host seven-time winners Aston Villa before Manchester City, who have also had seven triumphs, visit Bournemouth in the remaining last-eight ties.
**Reuters**