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The seventh seed won more than 80 percent of points on his first serve and clubbed 37 winners to bring an end to Gasquet's challenge after the ninth seed had battled back to win the second set.
The 27-year-old wrapped up victory with another huge serve after 140 minutes to level up their career head-to-head at 4-4 and join another Frenchman, Jeremy Chardy, in the last eight.
Tsonga, who beat Gasquest at the same stage of the 2008 tournament on the way to his only grand slam final, will meet the winner of the contest between Roger Federer and Milos Raonic being played later on Monday.
