Gasquet nabbed the first break point of the match, which kept him in control of the court for the remainder of the opening set.
One break was enough to give the Frenchman a one-set lead, and he continued his dominance by breaking Berdych's serve at the first time of asking in the second set.
Gasquet stormed to a 3-0 lead, but Berdych fought back with a break of his own, but it was not enough to trouble the Frenchman, who secured another break to push on and double his lead.
Berdych battled against his opponent to hold serve in the opening game of the third set, but the 28-year-old suffered a break point in the third game to give Gasquet the edge.
It didn't take long for the Frenchman to wrap up the contest and put his nation's first win on the board.