Benzema's 25th-minute goal meant Real have come out on top in each of their six games since Christmas, a burst of form that has seen them close the gap to just one point behind Barcelona and Atletico Madrid at the top of La Liga.
Espanyol have a good record against Real in the cup and knocked them out the last time the pair met in the 1999-2000 season, but Carlo Ancelotti's side put themselves in the driving seat with a solid performance.
It was a spirited display from Espanyol but it will take something special for them to turn the tie around at the Bernabeu.
The winners could play holders Atletico Madrid in the semis if they come through their tie with Athletic Bilbao while Barca, on the other side of the draw, face Levante.
"Espanyol made it difficult especially at the end and this was a good result for us," full back Alvaro Arbeloa told reporters.
"They (Espanyol) knew that 1-0 was bad and so pushed forward hoping to score in front of their own fans but we were solid," Alvaro Arbeloa told reporters.
"This is the line to follow for us as if we can keep secure at the back then we know we will get chances in attack with the quality we have there.
"There is a good relationship in the dressing room with the coach and we need to keep this run going."
Ancelotti put out a strong squad with only Xabi Alonso rested in the middle of the pitch as they looked to repeat their victory over Espanyol earlier this month in the league.
While the visitors linked up well in possession, Espanyol worked hard and Sergio Garcia unsettled Sergio Ramos and Pepe with a battling performance.
Espanyol might have taken the lead when Garcia, formerly of Barcelona, pulled the ball back for Pizzi after 15 minutes but the Portuguese shot over.
Cristiano Ronaldo responded with a strike against the post as Real began to put pressure on Espanyol and they went ahead when Arbeloa's cross was flicked on by Gareth Bale for Benzema to head home.
Arbeloa and Benzema failed to make the most of clear openings for Real after the break but the visitors needed Iker Casillas to make a smart save to deny Jhon Cordoba at the other end.
Image: Real Madrid's Cristiano Ronaldo and Karim Benzema
Photograph: Gustau Nacarino/Reuters
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