Rayo started the second leg with a 2-1 deficit, and their hopes of a fightback had looked to have disappeared when Paco Alcacer lost his marker in the eighth minute before slotting past Christian Alvarez.
Moments later, Rayo took the lead when Alejandro Pozuelo unleashed a low drive that nestled in the bottom corner of Rodriguez's net.
Los Vallecanos thought they had added a third on 35 minutes when a Pozuelo through-ball found Jonathan Pereira and the Villarreal loanee produced a neat finish only to be denied by the offside flag.
Although replays show the decision to be harsh, the speedy forward wouldn't be kept off the scoresheet for long as he steered a low strike off the post and past a helpless Rodriguez.
The first-half display saw Nuno Espirito Santo make three changes at half time and the substitutions had an instant impact as Valencia scored within 60 seconds of the restart through Morcillo's own goal, which levelled the tie on aggregate.
Los Che's early goal galvanised the home support in the Mestalla and cries for a penalty soon after were waved away as Rayo struggled to settle into their rhythm.
Rayo made it 4-2 on the hour mark thanks to Embarba's well-taken finish, but Alcacer netted his second on 65 minutes to give Valencia hope of sneaking through.
The home side then took the lead in the tie thanks to Rodrigo's strike on 71 minutes.
Rodrigo then put his side under late pressure after he was given a red card for kicking out at Morcillo, but Valencia hung on to progress.