Sergio Garcia bagged a hat-trick for the Periquitos with Christian Stuani completing the rout after Saul Niguez had netted an 81st-minute consolation for Rayo, who had goalkeeper Ruben Martinez sent off in the 25th minute.
The result ended a run of four winless matches for Espanyol, moving them up the table from 12th after taking just four points from their previous eight fixtures.
And Aguirre hopes his side can continue to improve and get some more points on the board.
He told rcdespanyol.com: "There are always things that can be done better....but the three points are a step forward.
"I have always felt calm for the players and see them working hard every day, every workout, even though recent results have not gone our way.
"Victory is a lift."
The Mexican also praised returning defender Victor Alvarez, who played 62 minutes at Rayo in his first appearance of the season following a heart operation.
"I said that lightning is impossible to hold him off," Aguirre said of the 20-year-old before praising his side's efforts on Sunday. "Our idea was to make transitions with speed and we succeeded in overcoming their game because they really are very good carrying the ball."