Gasquet, who defeated Andy Murray in the opening round, set up a semi-final clash on Saturday with Belgian 16th seed David Goffin.
Goffin advanced when his quarter-final opponent, Japan's Yoshihito Nishioka, pulled out before their match on Friday due to illness.
Gasquet underwent groin surgery in January and returned to competition in May.
On the women’s side, top seed Ash Barty rallied from a set down to power into the semi-finals of the Cincinnati Masters.
The Australian world number two turned dominant after losing the opening set, rushing to a 5-7 6-2 6-0 victory over Greek Maria Sakkari.
Barty will face either Czech Karolina Pliskova or Russia's Svetlana Kuznetsova in the semi-finals.
Japanese world number one Naomi Osaka retired due to injury from her quarter-final match with American Sofia Kenin.
Kenin was leading 6-4 1-6 2-0 when Osaka retired.
