The hosts at Cornella-El-Prat went a man down after 30 minutes when defender Victor Alvarez was sent off for a late challenge on Granada striker Youssef El-Arabi.
El-Arabi was then denied his ninth goal of the campaign as home goalkeeper Francisco Casilla produced a smart save to keep out the Moroccan's powerful drive.
Uruguayan striker Christian Stuani then went close for Espanyol nine minutes before the interval, his stinging shot well saved by Roberto following an excellent throughball from Sergio Garcia.
Espanyol would have fallen behind on the hour mark if not for the woodwork, El-Arabi again providing the threat for Granada when his close-range effort crashed against the crossbar.
The winner finally arrived 11 minutes from time, centre-back Moreno heading home Garcia's corner at the near post to secure an eighth win of the season for Javier Aguirre's men.
Granada are now winless in four league games and are just four points above the drop zone, while Espanyol move into eighth, nine points clear of danger.
