A threat at the opposite end early in the game, Roco was responsible for gifting a penalty to the hosts when he pulled down Javi Guerra as the Rayo forward looked set to convert a cross from close range.
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Roberto Trashorras stepped up from 12 yards and fired the penalty into the bottom corner as Rayo took control of the game.
The home side grew in confidence after breaking the deadlock, but another clear opportunity did not arrive until Lass Bangoura's effort at the front post was charged down by two defenders early in the second half.
Espanyol should have got back on level terms when Felipe Caicedo lost his marker to find space in the box, but Tono came to the rescue once again to get a hand to his low strike.
However, their hopes of avoiding defeat were dealt a blow when Victor Sanchez was shown his second yellow card with over 20 minutes remaining.
Rayo capitalised to double their lead through Guerra, who slotted home from close range after Bebe had beaten two defenders to find room for the cross.
There was still time for the hosts to wrap up a comfortable win two minutes later as Guerra took advantage of space on the edge of the box to score his second of the night with an emphatic finish.
Espanyol could have scored a consolation in stoppage time after Tito was shown a second yellow card for handling the ball in the box, but Burgui was unable to take advantage as Tono guessed correctly to keep out the penalty.