The hosts started the contest on a brighter note as Pizzi drove through two challenges and tried to curl home an effort in the sixth minute, but his shot was deflected high and wide of the goal.
Atletico soon started to have a good spell but the hosts held their own at Estadi Cornella-El Prat.
David Villa had a good chance on the half-hour mark when he collected a loose ball inside the Espanyol penalty box, but his final effort was nowhere near target.
Koke was unlucky not to put Atletico in the lead as he saw his effort come back off the post five minutes from the half-time interval.
Diego Costa has a decent chance to break the deadlock for the Madrid outfit in the opening minutes of the second half, but Francisco Casilla collected the ball under pressure between the Espanyol sticks.
Mario Suarez thought that he had restored parity on 69 minutes when he headed past Casilla, but he was adjudged to be in an offside position.
Atletico pressed on for the equaliser but failed to score, losing their winning run in the Spanish first division in the process.