Last year's semi-finalist Gasquet moved stealthily into the last 16 for the seventh time in his career on Sunday, finishing off Spain's Albert Ramos 2-6 7-6(5) 6-2 6-3.
The stylish Frenchman required only 31 minutes to seal victory on Court 18, having moved two sets to one ahead the previous evening against the world number 36.
His win was his 28th at Wimbledon, taking him to equal fifth on the all-time list for Frenchmen, level with Rene Lacoste.
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Gasquet and Tsonga have shared the spoils in their eight tour match-ups, but Gasquet, 30, had the better of his friend in the Wimbledon fourth round in 2007 and the 12th seed will need to recover from his marathon tussle against John Isner.
Tonga's match against the American lasted four hours 25 minutes, having been carried over from the fading light of Saturday with Tsonga two sets to one down overnight.
"One of my good friends. We know each other so well. We played since we were children," Gasquet told reporters of Tsonga. "I practised a lot with him, so of course it will be a different match.
"But we have to forget it for some hours and then life goes on."
Speaking to the media after the longest match of the tournament so far, Tsonga, 31, said: "I hope I will have a good recovery from this one and tomorrow be fit to play."
Gasquet, who also reached the semis in 2007, dismissed the suggestion of an easier route through after top seed Novak Djokovic was beaten on Saturday because they were in opposite halves of the draw.
"In the draw sometimes you need to be a little bit lucky. Sometimes a big seed is losing. So that's the case with Novak," he said.
"Now another player will go in some semis and final. You never know. So maybe Federer is not facing Djokovic maybe in semis, so it can be a different tournament."
(Reporting by Martyn Herman and Alison Williams; Editing by Ed Osmond)
