The winger converted a penalty only to then collect a second booking moments later, after Paco Alcacer and Andre Gomez had earlier put the visitors in cruise control
Paco Alcacer opened the scoring after just seven minutes of play with a sublime sliding finish before teeing up Andre Gomes' for the visitors' second midway through the first half.
Rodrigo then added a third in the second half from the penalty spot only to then get himself sent off just moments later.
However, so poor were Getafe from start to finish, that even with the extra man, they failed dismally to trouble a young and dynamic visiting side.
Valencia had come into the game on the back of two successive wins, over Espanyol and Malaga, and they effectively had this game wrapped up by the 20-minute mark.
Unsurprisingly, the in-form Alcacer proved pivotal. After some desperately sloppy play from Getafe's Sammir in the middle of the park, the newly-capped Spain international prodded the ball into the path of team-mate Rodrigo.
The winger's immediate return pass was slightly ahead of Alacacer but the striker made light of an ever-decreasing angle by ingeniously going to ground to slide across the turf and whip the ball into the far corner in one fluid movement.
Alcacer turned provided 20 minutes in. After a sublime first touch had taken Rodrigo away from Sergio Escudero wide on the right touchline, the ball was worked infield to Andre Gomes , who played a brilliant one-two with Alcacer before coolly slotting past a dreadfully exposed Vicente Guaita.
Having been involved in the first two goals, it seemed only fitting that Rodrigo got himself on the scoresheet in the 72nd minute.
Naldo needlessly tripped Sofiane Feghouli in the area while the midfielder was actually moving away from goal and the referee had no hesitation in pointing to the spot.
Rodrigo stepped up to take the spot-kick and calmly stroke the ball down the middle of the goal.
However, his sense of celebration quickly turned to embarrassment, with the winger dismissed for a second bookable offence after stupidly taking Escudero's legs from under him.
It made no difference to the outcome of the game, of course, with Valencia cruising through the remaining 18 minutes to draw to within two points of leaders Barcelona.