Monfils will play for a semi-final berth against the winner of a later fourth-round match between Serbian third seed Novak Djokovic and US 19th seed Mardy Fish. Monfils is 0-4 all-time against Djokovic and 0-1 versus Fish.
Monfils, seeded 17th, reached only his third career Grand Slam quarter-final after a 2008 semi-final run and 2009 quarter-final run at the French Open, each time losing to Roger Federer, a possible US Open semi-final foe.
"For me it's amazing to be in front of the American crowd in the quarters," Monfils said. "I feel really comfortable here, maybe (enough) to win here one day."
Monfils fired 14 aces and 40 winners to advance after two hours and 35 minutes over one of his best pals.
"It's not that easy," Monfils said. "Richard and I are very good friends. It was a great match."
Gasquet double faulted Monfils the only break of the first set in the third game and surrendered a decisive break to Monfils in the penultimate game of the second set.
A forehand winner brought Gasquet a break for a 4-2 lead in the third set after he had denied Monfils on six earlier break-point chances. But after he held to 5-2, Monfils roared through the last five games for the victory.
"I tried just to compete a bit," Monfils said. "I know it's tough to close the match. I know he got a bit nervous and he missed a couple easy shots. The key I think was to put one more in than him."
The only better Grand Slam finish for Gasquet was a 2007 Wimbledon semi-final run which was ended by Federer.