Monaco, from Argentina, beat the Spaniard 7-5, 6-3.
Williams had a harder-than-expected struggle against Spanish qualifier Laura Pous Tio, finally winning 4-6, 6-3, 7-5. Pous Tios is ranked No. 399 and Williams is No. 5.
Story continues below ↓advertisement | your ad hereVerdasco, who beat Andy Roddick in an ATP final two weeks ago in San Jose, Calif., said he had not adjusted to clay after not having played on the surface in seven months.
"I played better than the first two days here, but I still didn't have the level I needed to beat a solid player on clay - and one who played well like Juan," Verdasco said. "There is not much more to say."
Monaco credited his serve.
"I have beaten a top-10 guy who has been playing well for several years," Monaco said. "But I can't think about it too much because tomorrow I have to start from zero."
Fernando Gonzalez of Chile rallied to beat Eduardo Schwank of Argentina 3-6, 6-3, 7-5, while third-seeded David Ferrer of Spain defeated Uruguay's Pablo Cuevas 7-5, 6-4.
"When I won the first set, the second was easier and I played with more confidence and was more aggressive," he said.
In the other quarterfinal, No. 4 Juan Carlos Ferrero beat two-time defending champion and fellow Spaniard Nicolas Almagro 6-1, 7-5, 6-2.
In the semifinals, Ferrer will play Gonzalez, and Ferrero will face Monaco.
Williams got herself into several holes with loose ground strokes and showed this is her first tournament of the season on clay - her weakest surface.
"I just thought I'm not going to give it to her. That's the first thing I thought," Williams said.
Williams, who won here last year, was cheered on by thousands of fans.
"I really have never had that kind of support, ever in my life," Williams added. "Never, never at home, never anywhere. ... Never Fed Cup, never even the Olympics. ... It was just really amazing. I thought, OK, yeah, I can do this."
"If I never experience that again, I can remember today."
In other women's quarterfinals, Polona Hercog of Slovenia beat Agnes Szavay when the Hungarian player retired because of a left thigh injury. Hercog won the first set 6-4 and Szavay was up 1-0 in the second when she stopped playing.
In the other quarterfinals, No. 5 Carla Suarez Navarro of Spain beat No. 3 Gisela Dulko of Argentina 6-1, 7-6 (4), and Edina Gallovits of Romania edged Sharon Fichman of Canada 7-5, 6-2.
In the semifinals, Williams will play Gallovits, and Suarez Navarro will face Hercog.
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