Second seed Gauff was the highest-ranked player remaining, following Iga Swiatek’s surprise defeat by Yulia Putintseva on Saturday.
But the 20-year-old followed the world number one out of the door at SW19 after falling to a 6-4 6-3 defeat by 19th seed Navarro.
It meant there were just two top-10 players remaining in the women’s draw – fourth seed Elena Rybakina and Jasmine Paolini, who is seeded seventh.
Wimbledon has been Gauff’s toughest Grand Slam to crack having never gone past the fourth round.
She has made it to the semi-finals of the Australian Open and Roland Garros, while she won her first major at last year’s US Open.
Navarro will play Italian Paolini, who is also into the Wimbledon last eight for the first time after a tearful Madison Keys retired with an injury in the deciding set of their match.
Navarro outplays Gauff for biggest career win
Gauff had breezed through the first three rounds at this year’s Championships, underlining her title credentials with commanding wins.
But in her first test against a player in the world’s top 20, unforced errors came to the fore while she struggled to deal with the variety of Navarro’s game.
The pair looked evenly matched in the first set, trading breaks before Navarro struck a decisive blow with a forehand winner at 5-4.
Despite being cheered on with chants of “let’s go Coco, let’s go”, Gauff couldn’t find a way back after her opponent broke for a 3-1 lead in the second.
“I don’t have a ton of words but just really grateful to be out here on Centre Court,” said Navarro, who sealed the biggest win of her career on her third match point.
“A tournament with so much history and tradition that so many legends have played on before – it’s a real honour.”
The win continued a breakthrough season for the Navarro, who is into the last eight of a Grand Slam for the first time.
The 23-year-old won her first WTA title at the Hobart International in January, while she also reached the fourth round of a Grand Slam for the first time at last month’s French Open.
