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Fifth seed Tsonga won 7-5, 6-4, 6-4 against the 45th-ranked Fognini and could meet compatriot Gilles Simon for a place in the last eight should the 11th seed get past Stanislas Wawrinka, the 18th-seeded Swiss.
It was Tsonga's second win in two meetings with Fognini, who reached the quarter-finals last year only for a thigh injury to force him to withdraw and hand Novak Djokovic a walkover into the semi-finals.
"I have had some difficult moments here, now I am happy to be having some good ones," said Tsonga, who has now matched his best performances of fourth round runs in 2010 and 2011.
Seventh-seeded Czech Tomas Berdych ended Kevin Anderson's hopes of becoming the first South African man to reach the last 16 of a Grand Slam for nine years in a third round slugfest.
The 31st seed from Johannesburg was attempting to match comptariot Wayne Ferreira who made the fourth round at the Australian Open in 2003.
But after taking a two sets to one lead, he ran out of steam in the final set, when he needed treatment on his left thigh at every changeover, and lost 6-4, 3-6, 6-7 (4/7), 6-4 6-4, eight minutes short of four hours.
Berdych, a semi-finalist in Paris in 2010, will next take on either ninth seed Juan Martin Del Potro of Argentina or Marin Cilic of Croatia for a place in the quarter-finals.
"When I played him (Anderson) in the first round in Madrid, he was the toughest player I faced until the semi-finals," said Berdych, who went on to be runner-up to Roger Federer in the Spanish capital.
"So I was expecting a tough one today. He is a player who is coming up and playing really well."
World number 98 Malek Jaziri wasted a great opportunity to become the first Tunisian man into a Grand Slam third round when he squandered three match points against Spanish 20th seed Marcel Granollers.
In a match held over from Tuesday, Granollers edged past Jaziri 7-6 (7/1), 3-6, 6-1, 3-6, 7-5 and will face France's Paul-Henri Mathieu for a place in the last 16.
Later Friday, top seed Djokovic faces French qualifier Nicolas Devilder, the world ranked 286, in a battle of the highest-ranked and the lowest-ranked players left in the draw.
Djokovic is bidding to become only the third man to hold all four Grand Slams at the same time and first since Rod Laver in 1969.
Third seed Federer, fresh from setting a new Grand Slam match wins record of 234, tackles Nicolas Mahut of France.
