Serving for the win at 5-3 in the third set, Nadal was broken for the second time in the match and he then failed to convert three straight match points when leading 6-3 in the ensuing tiebreaker.
That was the end of the Dutchman's resistance, though, as Nadal converted his fourth match point for a 6-4, 6-2, 7-6 (6) win on Centre Court. The Spaniard is playing his first grass-court tournament since 2019, when he lost to Roger Federer in the Wimbledon semifinals. He is looking for his third Wimbledon title and has a chance at a calendar-year Grand Slam after winning the Australian Open and French Open to take his career tally to a record 22 major titles.
"To be in the quarterfinals here at Wimbledon after three years without playing here, it's amazing for me," Nadal said. "So very, very happy."
Nadal will next face 11th-seeded Taylor Fritz, the only American man left in the draw.
Nick Kyrgios, meanwhile, is returning to the Wimbledon quarterfinals for the first time since making his debut in 2014 after improving his record in five-set matches at the All England Club to 6-0. Kyrgios overcame an ailing shoulder and a fourth-set slump to beat 20-year-old American Brandon Nakashima 4-6, 6-4, 7-6 (2), 3-6, 6-2.
Kyrgios needed treatment on his right shoulder several times during the match and was broken two straight times in the fourth set.
But he raced out to a 5-1 lead in the fifth and converted his first match point with a forehand volley. Despite the troublesome shoulder, Kyrgios finished with 35 aces.
And there was little in terms of antics or arguments from the Australian, unlike his previous match against Stefanos Tsitsipas, when he repeatedly argued loudly with the chair umpire and was accused by his opponent of showing a lack of respect. Kyrgios also needed five sets to get past Paul Jubb in the first round.
Kyrgios earned fines of $10,000 for spitting in the direction of a heckling spectator at the end of that first-round match and $4,000 for an audible obscenity during his tempestuous win against Tsitsipas in the third round.
"I was able to just say, 'Wow, look how far I've come,' to myself. I was bouncing the ball before I served; I really just smiled to myself,'' said Kyrgios, owner of a tour-high 11 victories on grass this season. "I was like, 'We're here, we're competing at Wimbledon, putting in a good performance mentally.' It was rewarding.''
Fritz, after three of his countrymen lost in the fourth round, ended that streak by beating qualifier Jason Kubler 6-3, 6-1, 6-4 to reach the last eight of a Grand Slam for the first time.
"I'm glad I could get the win on the Fourth of July, being an American,'' Fritz said in an on-court interview.
It was Fritz's eighth straight victory on grass after winning a warm-up tournament in Eastbourne.
Garin came from two sets down and saved two match points in the fifth before beating de Minaur to reach his first Grand Slam quarterfinal.
The unseeded Chilean trailed 5-3 in the first-to-10 fifth-set tiebreaker before winning six straight points to take a 9-5 lead. He then converted his second match point to advance 2-6, 5-7, 7-6 (3), 6-4, 7-6 (6) after 4 hours, 34 minutes.
He is the first Chilean man to reach a Grand Slam quarterfinal since Fernando Gonzalez in 2009.
The 19th-seeded De Minaur had never lost a set against Garin in three previous meetings and looked set to maintain that streak when he led 3-1 in the third-set tiebreaker. The Australian then held two match points at 5-4 in the fifth.
The Associated Press contributed to this report.