Monfils was 3-1 up in a second-set tie-break before rain stopped play on Thursday, but returned 24 hours later to win 4-6 7-6 (7-5) 6-4.
Once play resumed on Friday, Monfils continued where he left off but squandered three set points before eventually taking the tie this distance.
Last year's runner-up Alcaraz struggled in the opening exchanges, with his French opponent finding a decisive break in the third game as Monfils took control of proceedings in Ohio.
And Monfils' experience ultimately told, as the 37-year-old held out for the remainder of his service games to set up a meeting with Holger Rune in the last 16, condemning Alcaraz to a shock exit.
World number one Jannik Sinner, meanwhile, had no need to take to the court on his 23rd birthday, as he was handed a walkover following Jordan Thompson's withdrawal.
Data Debrief: The old head prevails
Monfils is the oldest player to defeat an opponent ranked in the ATP top-three since Jimmy Connors defeated Michael Stich in Memphis in 1992.
The world number 46 starred on serve, firing 15 aces throughout the contest compared to Alcaraz's seven, while also clawing back six of the seven break points he faced.
The Spaniard started well, racing through the first set before rain delayed proceedings with the second set level at 6-6.
When play resumed 24 hours later, Monfils picked up where he left off, winning the tie-break and the final set, to book a place in the last 16 at Alcaraz's expense.
After struggling to find his footing, Alcaraz says he found it hard to adapt while playing on Centre Court.
"It was really, really tough for me. I felt like it was the worst match that I've ever played in my career. Couldn't play, honestly," Alcaraz said.
"I've been practising really well here in this tournament. The previous days, I was feeling great, hitting the ball clear, you know, moving well.
"Probably I felt like it was another sport playing on the Centre Court [than] on the other courts. The balls were faster on the Centre Court bounced much, much, more than other courts.
"I don't know how I felt like this way, but, honestly, I couldn't control myself. I couldn't be better. So, this match it was impossible to win, and that's all."
Alcaraz's frustration was clear to see as he smashed his racquet on the court four times midway through the third set.
"I felt sometimes that I wanted to break the racquet," Alcaraz added. "It never happened before, because I could control myself in those situations, in those feelings, and most of the time I could control myself, and it could go better in the matches or in the situation that I've been feeling before.
"[This time] I couldn't control myself because, as I said, I was feeling that I was not playing any kind of tennis. So, it was really frustrating for me.
"There is some point that I want to leave the court. I don't want to be on the court anymore.
"So, it was really hurtful moment for me, and that's it. I think some players or a lot of players during their careers and during some certain moments, they can't control themselves, and it was one of those moments for me."
Alcaraz will be chasing his third grand slam title of the year at the US Open, which starts on August 26 in New York.
The world number one took one hour and 37 minutes to earn his 7-6 (7-4) 6-2 victory, having battled through back-to-back three-setters in order to reach the final.
Sinner has been hampered by a hip injury, picked up during the Madrid Open in May, but rallied to get past the Frenchman.
"It was a very difficult week, tough week. I'm very happy about today's match," Sinner said in his on-court interview.
"It was very tough mentally. Had such an amazing run here, and I tried to do my best today.
"We both were quite tired from yesterday. We both felt a lot of tension, but I'm very glad about [the] level I played today, especially in the important moments."
There was little to separate the two in the first set, as they matched each other all the way to a tie-break, with Sinner eventually pulling in front to take the lead.
The Italian then got a vital break in the first game of the second set, taking confidence as he edged in front.
Tiafoe kept up the fight though, and successfully defended match point three times in the penultimate game, but he could only delay the inevitable as Sinner served to love in the last to get his hands on the trophy.
"It has been a tough week, some ups and downs, which is normal to have," Sinner added. "But how I handled the very important moments in each match, I'm very happy.
"I'm very happy to be in a position where I am and just trying to keep going mentally, having this hunger to keep playing, and hopefully, I can show some good tennis also in New York."
Data Debrief: Golden touch
Sinner has won his 15th career ATP title from 19 finals, equalling Carlos Alcaraz and Lleyton Hewitt (19). Only Rafael Nadal (17) has won more from his first 19 ATP event finals in the Open Era.
It is also his third ATP Masters 1000 title on hard court. In the past decade, only three players have won more Masters titles on the surface - Novak Djokovic (17), Roger Federer (seven) and Daniil Medvedev (five).
The world number one recovered from losing the opening set to prevail 4-6 6-3 7-5 after two-and-a-half hours, setting up a mouth-watering last-four clash with Aryna Sabalenka.
In the first ever meeting between the players, Andreeva made a positive start as she broke her opponent in the third game, while saving two break points in the sixth and 10th to draw first blood.
That sparked Swiatek into life; a break in the opening game of the second set proved enough for the four-time French Open champion to level the contest.
The Pole broke again in the decider and, after Andreeva squandered the opportunity to break back at 6-5 by sending a backhand long, she won the next two points to advance to the semi-finals.
Data Debrief: A perfect 10 from 10 for stubborn Swiatek
Matching her performance from last year in Cincinnati, where she was beaten in the last four by Coco Gauff, Swiatek will hope to go the extra steps this time around.
The Pole maintained her perfect quarter-final record in 2024, having now won all 10 such matches, and is now 16-1 at this stage in WTA 1000 events, with her only defeat coming against Elena Rybakina by retirement in Rome last year.
With her 30th main-draw victory at a WTA 1000 this season, she becomes only the second player after Serena Williams (36 in 2013) to reach that tally in a single year.
And Swiatek has now reached the highest percentage of semi-finals since the format's introduction in 2009 (17 from 31, 54.8%).
