A superb free-kick by Nolito was enough to see off Javier Aguirre's side, who were the better team throughout the match but were unable to take their chances.
The game took a while to get going, with neither side able to create many openings. Javi Lopez had the first real opportunity of the game when he burst into the box down the right hand side, but his cross-shot went wide.
With both teams well organised in defence, the match required something special for the deadlock to be broken. That moment arrived eight minutes before half-time.
Granada were awarded a free-kick 30 yards from goal after Gabriel Torje was fouled by Victor Sanchez. Nolito struck the ball with his right foot and it flew over the wall and dipped down just under the bar, leaving Espanyol goalkeeper Kiko Casilla frozen on his line.
Espanyol could have drawn level right on half time, but Granada stopper Roberto Fernandez produced an excellent save to keep out Sergio Garcia's header.
The Catalans were an improved side in the second half and should have got an equaliser. Christian Stuani headed over the bar from the back post after a cross from Joan Verdu and later on raced into the box and delivered a dangerous low cross across the face of the goal, but the ball was eventually cleared by the Granada defence.
Moments later Stuani was presented with another opportunity, this time volleying towards goal after a ball over the top by Verdu, but Roberto was equal to it.
Granada could have put the game to bed in injury time when Odion Ighalo found himself through on goal with just Casilla to beat, but the goalkeeper got down low to make the save and deny the Nigerian.
The final whistle sounded to confirm Granada's first win in nine games, one that takes them up to 16th in the La Liga table.