The Frenchman was broken twice in the first set as he struggled for rhythm on his serve, allowing the 29-year-old to cruise to a 6-2 victory.
However, Gasquet hit back impressively by taking the attacking to Berdych with a series of shots on his forehand and he managed to break the Czech twice to level the match courtesy of a 6-3 win.
The third set proved to be a tight contest as both players found their best form as neither was able to crack their opponent. However, the number 12 seed found another gear to create his opportunity to break and seized the one chance to enable him to take the set 6-4.
Berdych's spirit seemed to be broken by his loss in the third as the Frenchman seized an early break and the sixth seed was not able to take advantage of the opportunities that came his way to hit back.
Gasquet secured his second break of the set and he served out for the victory to take the fourth 6-1 to book his place in the last eight of the competition.
The Frenchman will play Roger Federer in the quarter-finals.