Thomas Tuchel's side bounced back from their Champions League exit on Tuesday with two second-half goals after a nervy opening period where they struggled to find a way through Palace's defence.
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Loftus-Cheek scored his first goal in three years to break the deadlock on the 65th minute, while Mount secured the victory 10 minutes later with a tidy finish into the bottom-right corner.
Chelsea will face Liverpool, who beat Manchester City 3-2 in the first semifinal on Saturday, in the final on May 14 in a repeat of the Carabao Cup final in February, where Jurgen Klopp's side claimed the trophy on penalties.
"I think it took a little deflection but I'll take it... it's long overdue," Loftus-Cheek said of his goal. "It's been a tough road for me, with injuries and going out on loan.
"We want to get [Liverpool] back, simple as that. It was a fantastic game in the League Cup final and could have gone either way."
It was a nervy start as Chelsea dominated possession during the opening period but could not break through the Palace defence, with the south London side looking dangerous on the counterattack.
The best chance of the half came just before the break from Palace midfielder Cheikhou Kouyate, who volleyed a clearance from Marcos Alonso towards Edouard Mendy, forcing the Chelsea goalkeeper into a diving save.
Chelsea were much improved after half time, and came close when substitute Loftus-Cheek, who replaced a limping Mateo Kovacic after 24 minutes, sent a ball through Palace's defence for Timo Werner to run onto.
However, Chelsea's breakthrough came after 65 minutes when Kai Havertz's ball across goal was deflected out to Loftus-Cheek, whose first-time strike on the volley took a deflection past Jack Butland and into the net.
Mount doubled their lead shortly after, weaving through the Palace defence before slotting the ball past Butland into the corner.
Romelu Lukaku nearly made it three for Chelsea with a minute to go, but his effort bounced off the right post.
"It was not easy because we played three games in three different competitions," Tuchel said. "It is not always easy, very challenging physically, we are the team with the most minutes in Europe I think, it is also challenging mentally.
"I thought the performance was very controlled and very serious, we took care of their counterattacks. It was hard, the opponent changed system twice, you cannot lose focus for a minute."
"I am happy to be part of it again," he added about facing Liverpool in the final. "I cannot be more excited and more proud because it is a huge competition. We will be ready."
It will be Chelsea's fifth FA cup final in the past six seasons, with the side claiming the trophy once in 2018.