Hewitt and Guccione were previously undefeated in the Davis Cup as a doubles team, but Tsonga and Gasquet overpowered them by hitting 21 aces to 4, and 74 winners to 38.
The Australians rallied from a 3-0 deficit and won the opening set when Hewitt hit a return winner off Gasquet's serve. However, Guccione failed to serve out the second set at 5-3 as the match turned in France's favor.
"If we could have gone up two sets to love, it could have been a different story," Hewitt said.
In the tiebreaker, the French team jumped to a 6-3 lead after Guccione netted a backhand volley and Tsonga slapped a forehand winner. Tsonga fired an ace to level at one set apiece.
The French pair did not face any break points in the third and fourth sets. The hosts went a set up when Gasquet whipped a forehand pass down the middle and won the match when Gasquet hit a powerful backhand that Hewitt could not return.
"It's a feeling of accomplishment," Tsonga said. "We played three good matches. We played unbelievable again today."
On Friday, Tsonga cruised past Hewitt 6-3, 6-2, 7-6 (2), and Gasquet routed Nick Kyrgios 7-6 (3), 6-2, 6-2.
France will face Germany in the quarterfinals while Australia will compete in the playoffs to stay in the World Group.