The sixth-seeded Tsonga, who won the indoor tournament in Marseille two years ago, dropped his serve once in the first set but didn't face another break point.
"I didn't really know how to play against him. I was a bit lost as to how to approach the whole thing," Tsonga said. "I pulled myself together, which is the most important thing. The fans helped me."
Meanwhile, fourth-seeded Jurgen Melzer of Austria had an easier path into the last eight, beating Andreas Seppi of Italy 7-5, 6-3. He next plays either No. 5 Ivan Ljubicic of Croatia or Russia's Dmitry Tursunov.
Tsonga broke Stakhovsky's serve three times over the last two sets, while both players finished with 11 aces.
"When I got going things were better," Tsonga added. "Players like that are able to hang in there over a whole game, except that when there's the right type of player in front of them there comes a time when they crack."
Simon beat Russia's Nikolay Davydenko 7-6 (6), 6-4 in the first round.