Second seed Tsonga had a tough time in his opening set against another Frenchman, Edouard Roger-Vasselin, losing it in a tie-break.
But world number 10 Tsonga found his game thereafter, and finished off the match in relative ease to win 6-7(3) 6-2 6-2.
Tsonga faces Germany's Jan-Lennard Struff in the semi-finals, while third seed Ernests Gulbis is the last-four opponent for Gasquet.
ATP Marseille Open quarter-finals
Jan-Lennard Struff (Germany) beat Michael Llodra (France) 6-4 6-3
2-Jo-Wilfried Tsonga (France) beat 6-Edouard Roger-Vasselin (France) 6-7(3) 6-2 6-2
3-Ernests Gulbis (Latvia) beat 8-Nicolas Mahut (France) 6-3 7-6(1)
ATP Delray Beach quarter-finals
Steve Johnson (U.S.) beat 6-Feliciano Lopez (Spain) 6-3 6-2
4-Kevin Anderson (South Africa) beat Marinko Matosevic (Australia) 6-7(3) 6-3 6-3