World number nine De Minaur won 6-2, 6-4, 4-6, 6-3 and will face Novak Djokovic for a place in the semi-finals if the seven-time champion defeats Holger Rune later Monday.
De Minaur appeared to roll his ankle as he played the winning shot on Court One and walked gingerly back to his courtside chair.
As he sat down, he shook his head towards his box, where 2002 Wimbledon champion Lleyton Hewitt, a fellow Australian, and tennis-playing girlfriend Katie Boulter looked on.
"I'll be all right," De Minaur said in his on-court interview.
"I'm super-proud to play in front of you guys. I had a few problems getting over the finish line and made it harder than it should have been."
World number 34 Fils, playing in the fourth round of a Slam for the first time, was undone by 66 unforced errors.
De Minaur has defeated both Djokovic and Rune in his career.
He got the better of the 24-time Grand Slam title winner at the United Cup in Australia this year while seeing off Rune twice in 2023, including on grass at Queen's Club.
"I'll try my best and play my heart out," said De Minaur, who is a force to be reckoned with on grass having clinched the 's-Hertogenbosch title in the Netherlands last month.
