The former finalist in Melbourne won 6-4 3-6 6-3 6-2 to advance to the last eight.
Tsonga raced into an early lead, powering the ball to the baseline before following it in with some superb volleying.
He won Gasquet's opening service game to love and followed it up with some big serving and great passing shots on both sides.
After saving a break-point in the third game, Gasquet got himself on the board with some good serving but Tsonga continued in the ascendancy with an exhibition of just about every tennis shot.
Gasquet began to show his own considerable skill set as the match went on, combining tremendous spin with intelligent angles to stay within one game.
The number nine seed eventually set up three break points when a wonderful backhand pass followed some excellent defensive work. Tsonga was able to serve his way out of trouble though and took the first set 6-4.
The second set began with Tsonga making more and more unforced errors, and when three in a row gave Gasquet a break, he capitalised on it to go 4-1 up.
Tsonga's serve was still quite strong but the damage had been done and Gasquet produced three set points with a forehand that curled away and on to the line, before taking the set 6-3 with an ace.
Two fortunate net chords went Tsonga's way in the second game of the third set. The number seven seed capitalised, finding a couple of fantastic backhands, and broke for 2-0.
Neither player was able to make any inroads into the other's serve thereafter, and after setting up the point with a 139mph ace, Tsonga took the third set 6-3.
He continued that form into the fourth set, breaking Gasquet in the first game, and again in third, as the number nine seed had little answer to his higher-ranked compatriot.
Gasquet did manage to get on the board in the fourth set, reminding the crowd of his sublime backhand in the process, but it was not enough and Tsonga served out the set at 6-2.