Gasquet battled to a 6-3, 7-6 (7-5) victory over Albert Montanes of Spain to make it into the semifinal round for the first time this year.
Argentine Juan Martin Del Potro eliminated German Philipp Kohschreiber 7-6 (7-3) 6-3 in another quarterfinal match.
Gasquet takes on Agustin Calleri of Argentina, who ousted German Michael Berrer 6-4, 6-2, in the semis Saturday.
In the other bracket, Del Potro goes against countryman Eduardo Schwank, who topped Czech Jan Hernych 6-0, 4-6, 6-3.
Celebrity Families of 2014 [PHOTOS]