The second-seeded Frenchman broke twice in the opening set and three times in the second to close out a commanding performance on clay in 66 minutes.
Gasquet, who lost to Novak Djokovic in the 2007 Estoril final, hit 24 winners.
The big-serving Swiss saved break points in his first two service games before being broken in the sixth game on a double-fault. He broke back immediately and had five set points at one stage in the tiebreaker.
Wawrinka overpowered his opponent in the 20-minute second set, serving out the match in damp and windy conditions at the rain-disrupted event.
In the women's event, top-seeded Roberta Vinci advanced to the quarterfinals with a 6-2, 6-0 victory over Maria Joao Koheler.