Gasquet, who has dropped to 15th in the ATP rankings, pressured Giraldo's serve throughout, but could only take two of his seven breakpoint chances and took more than two hours to beat the Colombian, who is ranked 172nd.
Nicolas Lapentti of Ecuador also won Tuesday, beating sixth-seeded Ivo Karlovic of Croatia 7-6 (4), 6-3, and Fabrice Santoro progressed with a 6-4, 6-1 win over Fabio Fognini.
Later, top seed Andy Roddick of the United States plays Frenchman Nicolas Mahut, and third seed Jo-Wilfried Tsonga takes on Marc Gicquel in an all-French encounter.
Ballon d’Or Nominees Announced
Moya, Simon reach final in Bucharest
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