Alcaraz was beaten 4-6 7-6 (7-5) 6-4 by France’s Monfils in a second-round match carried over from the previous day because of rain.
The 21-year-old Spaniard, who has won the French Open and Wimbledon titles this year, smashed his racquet after failing to break Monfils, 37, in the third set.
“I felt sometimes that I wanted to break the racquet,” said Alcaraz immediately after the defeat. “It never happened before, because I could control myself in those situations, in those feelings.
“Today, I couldn’t control myself, because, as I said, I was feeling that I was not playing any kind of tennis.”
The Olympic silver-medallist later apologised for his behaviour on his official Instagram account, external and said he would work on ensuring there would be no repeat at the US Open, which starts on 26 August.
“I want to apologise for my attitude yesterday, it was not good and that should not happen on court,” he wrote.
“I’m human, the nerves had accumulated and sometimes it’s very tough to control when there is that tension.
“I will work so that it doesn’t happen again. Now it’s time to think about NYC!”
Monfils went on to play Rune and was beaten 3-6 6-3 6-4 by the Dane.
Meanwhile, women’s world number one Iga Swiatek made it into the Cincinnati Open quarter-finals with a comfortable 6-2 6-2 win against Marta Kostyuk.
She will play Mirra Andreeva, who beat Italy’s Wimbledon finalist Jasmine Paolini 3-6 6-3 6-2.
“I’m happy I was solid. I was disciplined with all the things that I wanted to take care of,” said Swiatek.
Aryna Sabalenka secured a 7-5 6-2 victory against Elina Svitolina, while Jessica Pegula won a delayed second-round match against Karolina Muchova 5-7 6-4 6-2 before beating Taylor Townsend 6-2 6-3 in her second match of the day.