Neither Valencia, who are at the fifth spot in LaLiga, nor 10th-place Granada made many changes on Tuesday evening from the lineups they fielded in their respective league matches last weekend, reports Efe news.
Granada started brightly and took the lead in the third minute on a great individual effort by Roberto Soldado, who made up for flubbing a clear opportunity moments earlier.
The opening 30 minutes saw furious action at both ends.
For the hosts, Dimitri Foulquier and Yangel Herrera hit the post, while Alvaro Vadillo and Antonio Puertas tested Valencia goalkeeper Jaume Domenech.
Dani Parejo and Rodrigo spearheaded the Valencia attack, though the opportunities went to Maxi Gomez, who fired wide on one occasion and was bested by Granada keeper Aaron Escandell on a subsequent chance.
Valencia celebrated an equaliser by Rodrigo before VAR intervened, overruling the referee to disallow the goal for an offside.
Rodrigo persisted, however, and got his reward in the 40th minute in the form of an easy goal off a cross from Francis Coquelin.
Both sides were much better at the back in the second half and attackers had to work harder to find openings. Soldado missed the target before Aaron denied Maxi Gomez and Ferran Torres, but Domenech made the save of the night against Darwin Machis.
Granada looked the stronger team in the final phase of regulation and Domenech was called upon again to stop Machis, while Gonçalo Guedes wasted Valencia's best chance.
The teams were three minutes into stoppage time and looked headed for extra time when VAR detected a hand ball by Valencia's Jaume Costa on a corner and the referee pointed to the spot.
Rodrigo converted to eliminate the defending Copa del Rey champions and put Granada in the final four for the first time in more than half-a-century.
Second-division club Mirandes will host Villarreal in the second quarterfinal on Wednesday.
The last two semifinalists will emerge from Thursday's matches: Real Madrid-Real Sociedad and Athletic Bilbao-Barcelona.
( With inputs from IANS )