The Frenchman took the opening set 6-3 with a series of impressive breaks of serve that took the sting out of Tipsarevic's game.
The number nine seed failed to recover from the opening set's disappointment and looked in disarray as Gasquet piled on the pressure. The Serbian saw his serve broken again as Gasquet took the second set 6-3.
Gasquet came into the third set with an increased sense of urgency as he looked to finish off the tie. Tipsarevic had no answer for the 17th seed's superb backhand as he raced into a 4-0 lead. Gasquet broke serve again and then served for the match himself to claim a 6-1 win.
Gasquet now heads into the fourth round for a tie against David Ferrer.