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Sam Querrey is the final American remaining in the men's draw as Fish exited a day after the third-round ouster of John Isner by Mikhail Youzhny.
Djokovic stands 4-0 against his quarters opponent Monfils, whom he last beat in November's Paris Bercy final.
"It's very encouraging that at this stage of the tournament that I'm raising my game and feeling more comfortable on the court," said the winner. "I played a great match today.
"I really just try to focus on my game. I'm happy with the way things are moving on for me at this point."
The Serb reached a record sixth straight Grand Slam quarterfinal, 13th in his last 15 majors.
Monfils struck 14 aces and 40 winners as he broke Gasquet on five of his 17 chances.
In the women's draw, top seed Caroline Wozniacki dispatched Maria Sharapova, upending the three-time grand Slam champion, with the Danish winner claiming her 18th victory from 19 matches since Wimbledon.
Last year's losing finalist to Kim Clijsters, she has won 13 straight matches and lost just 10 games so far during the fortnight from four matches.
"I knew I could win, but I knew it was going to be tough," said the Dane. "I knew that I had to fight for every point, she wasn't going to go away. She was going to hang in there and keep fighting till the end."
Dominika Cibulkova ended the hopes of 2004 winner Svetlana Kuznetsova as the former champion committed 42 unforced errors in a 7-5, 7-6 (7-4) loss.
Estonian unseeded Kaia Kanepi eliminated 2009 semi-finalist Yanina Wickmayer 0-6, 7-6 (6-2), 6-1.