Hampered by a series of injuries, Real's hopes of turning the tie around were dealt a huge blow when Danilo's own goal handed Celta to lead on the night just before half-time.
Ronaldo gave Madrid hope with his first goal in the Copa del Rey for two years.
However, Daniel Wass restored Celta's lead five minutes from time before Lucas Vazquez's stoppage time header at least prevented Madrid from suffering a third defeat in four games.
Barcelona take a 1-0 lead over Real Sociedad into the final quarter-final, second leg on Thursday.
"We are out and disappointed. Most of all I am disappointed because the boys played so well," said Madrid boss Zinedine Zidane.
"We lacked just a few minutes more or one move (to score the third goal), but it wasn't to be."
By contrast, Celta are now just three games away from a first ever major trophy.
"The players were capable of knocking out a giant like Real Madrid," said Celta boss Eduardo Berizzo.
Madrid faced a huge challenge to turn the tie around with Zidane missing seven first-team regulars through injury.
However, it was the visitors who enjoyed the better chances early on.
Celta 'keeper Sergio Alvarez made a stunning save to turn Ronaldo's header onto the bar before the World Player of the Year hit the post with the goal gaping on the rebound.
- Helpless Danilo -
===================
It was the second consecutive Cup game away from home that the much-criticised Brazilian had opened the scoring with an own goal.
Yet, just when the tie was beginning to look beyond the European champions, Ronaldo brought the game back to life with a stunning free-kick that flew into the far corner from 25 yards just after the hour mark.
Madrid came within inches of levelling on aggregate moments later when the normally deadly Sergio Ramos headed wide from point-blank range.
However, they were caught out at the back when Jozabed teed up Wass to slot into the far corner.
Once again Real responded as substitute Vazquez turned a flick-on past Alvarez as the game entered stoppage time.
But it was too little, too late as Real's hopes of a first ever treble were extinguished.
Earlier on Wednesday, Atletico secured their place in the last four despite resting star forward Antoine Griezmann with a 2-2 draw at Eibar to progress 5-2 on aggregate.
Disclaimer: No Business Standard Journalist was involved in creation of this content
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