Frustrating car the 36-year-old Frenchman wasted many chances. In the first set, he had five set points while serving at 5-3, but he converted none and even offered the break to his opponent by committing three consecutive double faults.
Van de Zandschulp pushed his advantage by winning the next three games to conclude with a white service game.
Then in the tie break of the third set, he committed another double fault at 5/3, on a point which could have allowed him to give himself three set points.
Instead, van de Zandschulp came back and won the set on his second set point, but not before dismissing a set point for Gasquet.
In the next round, the 26-year-old Dutchman who is participating in his second Wimbledon will face Rafael Nadal (4th) or Lorenzo Sonego (54th).
His best result in a Grand Slam tournament is the quarter-final played last year at the US Open for his first appearance.
For his part, Gasquet had not reached the third round of Wimbledon since 2016, where his road had stopped in the eighth.