No. 21 seed Gasquet, who had won his previous four matches against last year's Wimbledon semi-finalist, came through 6-3 6-4 6-4 on Centre Court to head into the second week at the All England Club for the first time since 2012.
Dimitrov was far from his best in front of a muted crowd, with his much admired backhand stuttering, and was too often outshone by what is also Gasquet's trademark shot, which left him rooted to the spot on several occasions.
The 24-year-old Bulgarian handed Gasquet a break in his first service game with a double fault and passed up chances to break back as the Frenchman settled into his stride quickly, punching shots into the corners to keep Dimitrov on the run.
The theme continued, with Gasquet eventually sealing the win at the fifth attempt when Dimitrov's return sank into the net.
While Dimitrov showed flashes of his talent the defeat will do little to silence critics who have questioned whether he can make the most of his potential.
He won three titles on three different surfaces in 2014 and reached the last four at Wimbledon but has drawn a blank since the turn of the year.
Gasquet, a Wimbledon semi-finalist in 2007, will next face Australia's 24th seed Nick Kyrgios who beat seventh-seeded Canadian Milos Ranoic in four sets.
(Editing by Ken Ferris)
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