Abdul Mumin, Andrei Ratiu and Unai Lopez scored the goals for Rayo on the night.
Perez said: "I think the match was divided into two parts. I sincerely believe that both are ours in terms of the search for victory. They scored a goal against us from something that they are dominating. It is a long ball, they win it and they can finish the play. At Osasuna it is historic and something that is always done.
"In the first half we were good enough for what is needed in a Primera Division match, especially at home. At half-time we tried to correct that and then make some tactical adjustments to improve in terms of the opportunities created."
On playing James Rodriguez from the bench, Perez explained: "Regarding James, I have said it many times, I am delighted that he is here. I think that the reception he is receiving from those involved has been excellent. You know what the dressing room is like, he is very happy but he is still just another Rayo Vallecano player.
"He comes from a fairly long period of inactivity. Even though he is already looking for minutes with his national team and with us. I make the changes based on what I think the team needs and not for other reasons. He will surely play more regularly."