The top four seeds have made it to the final eight with No. 3 Moya and No. 4 Robredo joining top-seeded Guillermo Canas of Argentina and second seed Younes El Aynaoui of Morocco.
Moya cruised past Irakli Labadze of Georgia, 6-3, 6-4, inching closer to his second title of the season. The former world No. 1 and 1998 French Open champion has eight career victories -- seven on clay.
Robredo defeated Spanish qualifier Gorka Fraile, 6-4, 2-6, 6-4. The 20-year-old has one title in four years on tour, winning on clay last season.
"I thought I played very well at the end of the third set," Robredo said. "Overall, I feel like I'm playing great tennis."
Norman has been slowed by a hip injury since finishing fourth in the ATP Champions race two years ago, winning just four of 11 matches this year. He claimed five tournaments and was the runner-up at the French Open in 2000.
"For the first time in a year, I feel comfortable and I feel good about my chances of winning a tournament," he said.
Canas will meet Albert Montanes in the quarterfinals. El Aynaoui will battle Bohdan Ulihrach of the Czech Republic.