The 31-year-old Gicquel fended off seven break points to beat Gasquet for the first time in three encounters and advance to the quarterfinals of the grass-court tuneup for Wimbledon.
He next plays Austria's Jurgen Melzer, who upset fifth-seeded Igor Andreev of Russia 6-4, 1-6, 6-1.
In the women's draw, defending champion Anna Chakvetadze of Russia defeated Angelique Kerber of Germany 6-3, 3-6, 7-6 (1).
AUSTRIA ORGANISERS CONSIDER CHANGES
French Open Show Court Schedules
