Four-time US Open champion Nadal, bidding for a 23rd Grand Slam title, lost 6-4, 4-6, 6-4, 6-3 in the last 16 to 26th-ranked Tiafoe.
The American's free-swinging performance conjured up 18 aces and 49 winners past a sluggish Nadal.
"I don't know what to say, I'm so happy. He's one of the greatest of all time and I played unbelievable tennis today," said Tiafoe who had only previously reached one Slam quarter-final in Australia in 2019.
"Something special happened today. Honestly when I first came on the scene I wasn't ready for it mentally and mature enough.
"I've been able to develop and I have a great team around me. With you guys, it's been great."
Tiafoe will face Andrey Rublev, who he defeated over five sets in the third round in 2021, for a place in the semi-finals.
Seventh seed Rublev shrugged off a rain delay to breeze past Cameron Norrie and reach the quarter-finals for the third time with a 6-4, 6-4, 6-4 win.
For Australian and French Open champion Nadal, it was a first Grand Slam loss of 2022 after an abdominal strain forced him to forfeit his Wimbledon semi-final.
His defeat came just a day after world number one and defending champion Daniil Medvedev was knocked out also in the fourth round by Nick Kyrgios.
"I wasn't able to push him back. Tennis is a sport of positions a lot of the time," said 36-year-old Nadal.
"You need to be very quick and very young. I am not in that moment anymore."
Nadal added: "I was not able to cause great damage. He was better than me."
- Alcaraz's determined victory -
Carlos Alcaraz defeated 2014 champion Marin Cilic 6-4, 3-6, 6-4, 4-6, 6-3 in a match which was just three minutes short of equalling the record for the latest finish at the tournament.
In a gruelling three-hour and 54-minute encounter, Cilic was under siege, having to save 12 of 18 break points while committing 66 unforced errors.
"I believed in myself, it was pretty tough," said Alcaraz.
"I was a break down in the fifth set. It was tough to come back after losing the fourth set."
The 19-year-old Alcaraz, seeded three and seen as the title favorite after the elimination of Nadal, will face Jannik Sinner of Italy for a semi-final place.
Italian 11th seed Sinner reached his third Slam quarter-final of 2022 with a 6-1, 5-7, 6-2, 4-6, 6-3 win over Ilya Ivashka of Belarus.
The pair served up 27 double faults between them and 119 unforced errors.
- WTA quarter-finals decided -
French Open champion Swiatek recovered from a set and a break down to see off 108th-ranked German opponent Jule Niemeier, 2-6, 6-4, 6-0.
In a mistake-riddled clash on Louis Armstrong Court, there were a total of 15 double faults, 12 breaks of serve and 76 unforced errors.
"It's really satisfying. This is my first quarter-final in New York so I am really proud of it," said Swiatek.
Wimbledon quarter-finalist Niemeier hadn't won a hardcourt match in her career before coming to New York.
However, she went into Monday's match not having dropped a set and fired up to repeat her Wimbledon journey where she had also beaten world number two Anett Kontaveit.
The 23-year-old from Dortmund swept through the opener and broke to lead 2-1 in the second set before Swiatek recovered to take control and reach her third Slam quarter-final of 2022.
Swiatek will face US eighth seed Jessica Pegula for a place in the semi-finals.
Pegula became the second American woman into the last eight after brushing aside two-time Wimbledon champion Petra Kvitova 6-3, 6-2.
World number six Aryna Sabalenka reached the quarter-finals for a second successive year with a 3-6, 6-3, 6-2 over Australian Open runner-up Danielle Collins.
Sabalenka has now beaten the American in New York three times in the last five US Opens.
The Belarusian will next face 2016 runner-up Karolina Pliskova who downed three-time finalist Victoria Azarenka 7-5, 6-7 (5/7), 6-2.
