Tsonga, who clinched a marathon match at the weekend against John Isner 19-17 in the final set, was 4-2 up in the first set against number seven seed Gasquet when the match was ended.
The 31-year-old will play British number two seed Andy Murray in the last eight - and said the tie held no fear for him after the epic Isner encounter.
"That's why tennis is great. Two days ago I was 5-5 in the third set, 15-40 against me, two sets to love down, and I came back. I'm still alive in this tournament," Tsonga, seeded 12, told reporters.
"Everything can happen in tennis. I'm very confident of my capacity to play great tennis and beat players like Andy."
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Gasquet said he had first felt pain in his back towards the end of his previous match, against Albert Ramos-Vinolas.
"This morning when I warmed up, it was okay. Then during my match, I felt something wrong and I couldn't move after that."
He said he was not sure exactly what kind of injury he had suffered, adding: "There are tests to come."
(Reporting by Pravin Char; editing by Ken Ferris)
