Florin Andone had put Cordoba ahead but Miroslav Dukic’s side conceded twice in the final five minutes and remain four points adrift at the foot of the Primera Division.
An own goal from Fede Vico had appeared to give visiting Getafe a share of the spoils before Rodriguez headed in a 92nd-minute winner to help their own survival hopes.
It was Getafe who started sharply and could have opened the scoring early on as Diego Castro’s well-struck effort flew behind for a corner after deflecting off Cordoba defender Eduard Campabadal.
Cordoba were also quick out of the blocks and had their own chance to take the lead shortly afterwards as Nabil Ghilas cut inside before arrowing an effort wide with Campabadal well-placed in the middle.
Pedro Leon was the next to come close for Pablo Franco’s Getafe but could only whistle his shot wide of Mikel Saizar’s goal after being picked out by Pablo Sarabia’s square ball.
Vicente Guaita made two great saves to keep the scores level as the hosts pushed for a half-time lead.
Firstly he kept out a Campabadal drive and moments later was on hand to prevent Bebe scoring from a free-kick.
After the break, Ghilas passed up two half chances before Rodriguez drew a save out of Saizar with neither side able to take a stranglehold on the game.
Adone came off the bench and headed a Bebe set-piece wide just after the hour-mark but, just a little over 15 minutes later, he was celebrating an opening goal.
The Romanian striker showed nimble footwork to work his way clear before flashing home an unstoppable effort that looked on course to give Cordoba a first league win since January 12.
But a horrible mix-up with just three minutes remaining saw Getafe pull level as Vico appeared to change his mind about hooking a loose ball clear before finally getting something on it and putting it past Saizer to score an own goal that will be remembered for all the wrong reasons.
That was not the end of the heartache for the home side as Rodriguez headed home Sarabia’s late corner to seal a late victory for Getafe and move them five points clear of the relegation zone.