The Portugal forward, runner-up to Lionel Messi in Monday's World Player of the Year awards, netted twice inside the first 25 minutes at the Bernabeu and sealed the victory near the end after Sergio Ramos had been sent off.
Real's 5-2 aggregate win set up a meeting with Valencia next week.
"It needs to be pointed out, Cristiano has scored four goals across the tie," Real director Emilio Butragueno told Spanish television.
"His contribution has been decisive. He has been the player of the tie, and we are lucky to have him."
The cup has taken on added importance for Jose Mourinho's Real with Barcelona leading them by 16 points in La Liga and, after losing 2-1 in Vigo, the pressure was on.
Mourinho fielded the strongest side available to him with Iker Casillas restored to the starting line-up after being dropped to the substitutes' bench for the last two games.
The return of the popular Spain number one led to cheers when his name was read out before kick-off, while Mourinho's was once again met with a mixture of whistles and applause.
"This is a stadium that is very demanding," Butragueno replied when asked about the whistles for Mourinho, who did not speak after the game.
"I remember the fans here getting on my back too, the most important thing was the team's performance, particularly in the first half."
Real's influential midfielder Xabi Alonso exited with a neck strain at the break, and his loss helped to breathe life into the visitors.
A lung-bursting sprint down the pitch enabled him to latch on to Gonzalo Higuain's pass and fire in his third in the 87th, and Khedira finished off another counter-attack a minute later.
