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."
Real Zaragoza were 2-0 winners at home to Levante with goals from Franco Zuculini and Francisco Montanes to progress 3-0 on aggregate, and set up a last-eight meeting with Sevilla.
The 2010 Cup winners Sevilla were never seriously troubled having won 5-0 away at Real Mallorca in their first leg, and in a sparsely populated Sanchez Pizjuan progressed 6-2 on aggregate.
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.
His stewardship of the club has come under increasing scrutiny in recent weeks from local media and disgruntled fans.
"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."
It was Celta who sprang the surprises, leaving regulars such as leading scorer Iago Aspas on the bench, and the hosts threatened to swamp the Galicians in the first half.
Ronaldo struck a swerving shot from wide on the touchline, which flew inside the far post after only two minutes to set the tone.
Luka Modric floated a ball through for Ronaldo to stroke in the second in the 25th and it was something of a surprise he did not have his hat-trick by half-time.
Real's influential midfielder Xabi Alonso exited with a neck strain at the break, and his loss helped to breathe life into the visitors.
Ramos picked up a second yellow card when he fouled Augusto Fernandez in the 73rd, and Casillas was forced into some sharp saves.
Ronaldo rode to the rescue again, as he had done with two second-half goals in the 4-3 win over Real Sociedad on Sunday.
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.
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