Medvedev, who underwent hernia surgery last month, cautioned Sunday he was often a slow starter on clay courts. So it proved in his first match on the surface this season, in the last tournament before the French Open.
Medvedev double-faulted for the seventh time on Gasquet's first match point, as the Frenchman ended a 36-match losing streak against opponents ranked No. 1 or 2 after beating Roger Federer 17 years ago at Monte Carlo.
“It's tough. I don't play my best tennis on clay courts," Medvedev said on Tuesday.
"I kind of need to be in the zone, which I was not today.”
Unlike Medvedev, Gasquet has won titles on clay in his career and the 35-year-old Frenchman frustrated his top-seeded Russian opponent in an assured first set.
Medvedev had three double-faults when his serve was broken to open the second set, and he broke a racket on the court in the next game. He leveled at 3-3 and clinched the next game at love with a second-service ace.
In the tiebreaker, Medvedev twice recovered from being a mini-break down before his serve faltered again.
“I think more days of practice and (I) should be ready for Roland Garros because physically, even right now, my body feels good,” he said.
Gasquet's will face Kamil Majchrzak in the quarterfinals after the Polish player beat qualifier Marco Cecchinato 6-2, 6-3. Cecchinato had beaten Dominic Thiem, the two-time French Open runner-up, on Monday.
In a first-round match, fifth-seeded Nikoloz Basilashvili advanced when Facundo Bagnis retired after losing the first set.