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Anderson, the seventh seed, won the first set against second seeded Verdasco, the former world No 7.
The Spaniard bounced back, however, to win the second set and held off a fighting Anderson to clinch the match.
Anderson was made to work hard in his first round match against Thiemo de Bakker of the Netherlands before brushing aside Romanian Victor Hanescu in straight sets to reach the final eight.
The 24-year-old, ranked 34th in the world, has had a superb season on the ATP Tour.
He reached the semifinals in Brisbane in early January where he was knocked out by former world No 1 Andy Roddick of the US, and four weeks later defeated India's Somdev Devvarman in the final of the SA Open in Johannesburg to win his first ATP 250 tour title.
In February he reached the quarterfinals at Delray Beach where he lost to former world No 4 Juan Martin del Potro of Argentina.
One month later Anderson stormed into the last eight of the ATP World Tour Masters 1000 event in Miami, where he was ousted by world No 2 Novak Djokovic of Serbia.
Anderson's performance in Estoril could see him continue his rapid climb up the world rankings, which will be released on Monday.
