Nadal, who received a bye as second seed, was playing his first singles match in two months after spending time at home in Mallorca with his wife and newborn son.
The 22-time Grand Slam champion shook off an early break in the opening set by winning five of the next six games.
But Paul recovered from dropping serve in the third game of the second set to win a tie-break, having failed to convert a set point with Nadal serving at 4-5.
The Spaniard's lack of match practice told as Paul dominated the final set, breaking three times to seal the biggest win of his career.
"I had the match (in my hands) in that second set with a set and break. I played a terrible game there. I didn't deserve the victory playing that bad in that key moment," said Nadal.
Nadal is a 14-time French Open champion but has never won the title at Bercy.
"A lot of things have been going on the last couple of months, without a doubt. But we are always ready to find excuses. At the end, it's always the same. You play well, you win; you don't, you lose," he added.
Nadal said he expected to compete at the Tour finals in Turin from November 13-20.
"If nothing happens, I hope to be there. I'm excited about playing, even if it hasn't been the perfect couple of months for me."
Paul goes on to play Pablo Carreno Busta for a place in the quarter-finals.
Alcaraz, the reigning US Open champion, will play Grigor Dimitrov in the last 16 after defeating Japan's Yoshihito Nishioka 6-4, 6-4.
The 19-year-old smacked 30 winners including nine aces despite a heavily-taped left knee, an issue he had downplayed on Monday.
"I tried to find my best level. I think I played really well. I don't think about the pressure of being world number one," said Alcaraz.
The Spaniard was presented with the world number one trophy earlier in the week, having become the youngest player to top the ATP rankings in September.
"I felt number one before getting the trophy, but of course with the trophy, is amazing. It was amazing feeling when I had the trophy in my hands."
Alcaraz admitted there is a target on his back now as the world's top-ranked player.
"Probably the difference, the way the players play against me, probably I felt that difference, but not too much," he said.
"I would say the players play better or more aggressive with me."
Nadal's loss means Alcaraz will end the season as the top-ranked player if he takes home the trophy on Sunday.
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