Paul stunned by Machac as Japan Open upsets keep coming, Rune advances
The American's departure means only two of the eight seeds are still standing in Tokyo.
Top seed Taylor Fritz, number three Casper Ruud, fourth seed Stefanos Tsitsipas and number seven Frances Tiafoe all lost in the first round.
Hubert Hurkacz, seeded two, was beaten by Britain's Jack Draper in the second round.
Denmark's number six seed Holger Rune avoided an upset with a 6-2, 6-4 win over Japan's Yoshihito Nishioka.
But Paul failed to join him in the quarter-finals after letting the Czech Republic's Machac back into a match that the American looked to have total control of.
"Even the not-seeded players here are great," said Machac, who has a world ranking of 34 and is looking to win his first career title.
"I'm just happy that I'm in and I just try to believe as much as possible."
Paul won five straight games in the first set but Machac came roaring back in the second and dug in when he found himself 3-0 down in the final set.
"I feel great," said Machac. "First set he was better, second set I found my rhythm. In the third set I was playing unbelievable so I'm super happy with my performance."
Machac's quarter-final opponent will be American Alex Michelsen, who beat Australia's Christopher O'Connell 6-1, 6-4.
Japan's Kei Nishikori also reached the quarter-finals with a 6-2, 6-3 win over Australia's Jordan Thompson.
Nishikori has been plagued by injuries and he returned to Grand Slam competition at the French Open in May after an absence of almost three years.
He was in inspired form as he beat Thompson in 1hr 22min but he refused to get carried away with his performance.
"It's just one match, I can't let my guard down," said Nishikori.
"If you only look at today's match it was really good, but the opponent is different every time so you can't judge someone on one match alone."
Nishikori is a two-time Grand Slam finalist who was once ranked number four in the world.
"I felt really good in today's match," said Nishikori, now ranked 200.
"That might be just because I was playing in Japan with the support of a lot of fans, so if I can keep this going overseas as well, it will give me more confidence."
Nishikori will face Rune in the quarter-finals after the Dane saw off Nishioka in a convincing performance.
"I just had one home player here against Nishioka and now it's Nishikori," said Rune.
"He's a great player, he's been great for many, many years.
"I watched him a lot when I was a kid so I'm looking forward to that match."