Gasquet, who is the higest seed left in the tournament following the exits of Marcos Baghdatis, Gaston Gaudio and Dominik Hrbaty, was joined in the last eight by compatriots Sebastien Grosjean and Arnaud Clement.
"I had to keep my concentration until match point," said French number one Gasquet.
"Now I have to face Robin Soderling who is one of the best players in the world on carpet. It will will be a very difficult game."
Clement, winner of the event in 2000 but now ranked 50th, defeated fellow Frenchman Michael Llodra 2-6, 6-1, 7-6.
Sixth seed Grosjean was a 7-6, 6-2 winner in another all-French encounter against Cyril Saulnier.
Grosjean, ranked 30th in the ATP standings, had to save a set point in the opening set before strolling through the second set.
"Cyril played a very good game. I think I was a bit lucky to break him when he was serving for the set at 6-5," said Grosjean, who will face compatriot Marc Gicquel in the quarter-finals.
"I played better afterwards and two passing-shots gave me the win."
Seventh seed Xavier Malisse of Belgium also reached the quarter-finals through a 6-2, 7-6 win over Frenchman Julien Benneteau.
He will play American Robby Ginepri, who crushed German Simon Greul 6-1, 6-2.
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