In a match that was suspended by rain on Thursday night with the players in a second-set tiebreaker, the 37-year-old Monfils advanced by taking the last two sets from the second-seeded Alcaraz, who called the loss his "worst match."
"I felt like it was the worst match that I've ever played on my career," Alcaraz said. "I've been practising really well. I was feeling great. But I couldn't play. I want to forget it and try to move on to New York."
Trailing 3-1 in the tiebreaker when the match was halted, Alcaraz was hoping for a reset when play resumed on Friday. But, the No.3 player in the ATP rankings wasn't able to control his emotions.
"It never happened before because I could control those feelings," Alcaraz said. "Today I couldn't. I was feeling that I was not playing any kind of tennis. It was really frustrating for me. At some point, I didn't want to be on the court anymore."
Alcaraz was playing his first match since earning a silver medal at the Paris Olympics after losing to Novak Djokovic, who also beat him last year in the Cincinnati final.
Monfils returned to the court later on Friday and lost to No.15 seed Holger Rune 3-6 6-3 6-4, as he tried to become the oldest men's quarter-finalist in Cincinnati in the professional era.
"It was a big win for me against Carlos, unexpected to be honest," Monfils said. "To play again in a couple hours against (another) good and younger player, I knew it would be tough."
The US Open begins in New York on August 26.
Top men to advance included No.1 Jannik Sinner, who got a walkover against Jordan Thompson on his 23rd birthday, No.4 Alex Zverev and No.6 Andrey Rublev.
Australian Thompson withdrew because of a rib injury.
