After losing last week's first leg 1-0, there seemed little hope of progression for the home side when Jorge Molina put Getafe 1-0 up inside a minute but Rodrigo's second-half treble turned the game on its head and sent them into the last four.
Molina, who got the winner in the first game, caught keeper Jaume Domenech out of position at his near post for the opener. However, Valencia showed the kind of spirit their La Liga season has been sorely lacking to fight back and progress.
The hosts thought they had equalised just after the break but the Video Assistant Referee showed that Carlos Soler was in an offside position before teeing up Santi Mina and it looked like it was going to be a frustrating night at the Mestalla.
However Rodrigo equalised on the hour and when Djene Dakonam was sent off for the second time in as many games for a second booking in the 74th, the pendulum swung in Valencia's favour as the game grew scrappy with 12 yellow cards handed out.
Substitute Hugo Duro missed a glorious chance to seal the tie for Getafe against the run of play before Rodrigo fired his second two minutes into added time for Marcelino Garcia's side.
Molina saw team mate Duro unintentionally block his goal-bound effort three minutes later, allowing Valencia to counter as Mina nodded the ball into the path of Rodrigo who scored the third to seal a famous victory and a place in the semis.
Tempers flared at the final whistle as players and coaches from both sides squared up.
In Wednesday's quarter-final second legs, Real Betis host Espanyol after a 1-1 draw in the first leg, while Barcelona look to overturn a 2-0 deficit against Sevilla at the Nou Camp before Real Madrid visit Girona with a 4-2 lead on Thursday.
(Reporting by Joseph Cassinelli; Editing by Ken Ferris)
