Frustrating because 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 3rd round of Wimbledon since 2016, where his road had stopped in the eighth.
He arrived at the All England Lawn Tennis Club this year unprepared for the turf.
Last year, he was eliminated in the second round by Roger Federer, but he reached the semi-finals there in 2015 (beaten by Novak Djokovic) and 2007 (beaten by Federer).
© 2022 AFP