Both players exchanged a break of serve in the opening set, but it was Granollers-Pujol who moved ahead in the match courtesy of a 7-2 score in the tie-break.
Gasquet recovered in the second, however, breaking his Spanish opponent at 1-1 before claiming the set 6-4.
Three breaks of serve occurred in the third set, but it was Gasquet who once more secured the vital game as he broke at 5-5 before serving for a two-sets-to-one lead.
The Frenchman took control in the opening match of the fourth set, breaking before moving 2-0 ahead.
Granollers-Pujol did recover to hold his next four service games, but it was not enough as Gasquet booked his spot in the second round with a comfortable service game.