The Spaniard looked to be heading for a decider until a brilliant second-set comeback saw him win 6-3 7-6 (9-7) on Court Philippe Chatrier in two hours and one minute.
Alcaraz was looking to quickly put his disappointing doubles exit with Rafael Nadal behind him and successfully defended two break points in the opening game.
While Paul matched him stride for stride during the opening stages, the Spaniard got the all-important break in game six to edge in front before taking the first set.
His momentum soon faltered though, as the American raced into a 3-0 lead in the second and looked to be cruising.
Alcaraz's fighting spirit soon came to the fore as he got another vital break during a three-game winning run to level the score and eventually, force a tie-break.
The 21-year-old then rounded off an incredible comeback by saving a set point during the tie-break and winning the final three points.
He will play either Norway's Casper Ruud or Canada's Felix Auger-Aliassime in the semi-finals.
Data Debrief: Alcaraz stays on track for gold
Alcaraz is the fourth-youngest player to reach the men's singles semi-finals at the Olympics since 1988 (21 years and 83 days), and the youngest since Novak Djokovic in Beijing 2008.
He has now won his last 11 matches on the bounce, winning each of the last four in straight sets.
Alcaraz beat Paul en route to his Wimbledon title last month, and he will be hoping that is a good omen as he chases a gold medal in his debut Olympics.
Murray's final tournament ended in the quarter-finals at Roland-Garros as he and Dan Evans were beaten in straight sets by American duo Tommy Paul and Taylor Fritz.
The 37-year-old left Court Suzanne-Lenglen to a standing ovation as the curtain came down on a career that included three Grand Slam titles, as well as singles gold at both the London and Rio Games.
"I feel good. I have been ready for this moment for the last few months since I had the operation on my back," said Murray.
"I knew it was the right time and I feel lucky to come and compete here.
"I am grateful to Dan (Evans) for putting in the effort he did to give us a good chance of having a great run. I am looking forward to stopping now."
While Murray and Evans had pulled off miraculous comebacks against Japan’s Kei Nishikori and Taro Daniel and Belgium's Sander Gille and Joran Vliegen in the previous rounds, both Paul and Fritz proved to be a step too far.
The British pair were second best for much of the contest, despite showing signs of yet another comeback in the second set only for Fritz's strong service game to cinch victory for the Americans.
"I mean it was a great run and we lost to two top players today. They are both top tennis players, we would have had to play very well to win that match and we didn’t.
"We got into it at the end, but if we found that form earlier we could have caused them problems, but it wasn't to be.
"We did make the quarters, but if you get through that match you get an opportunity to win a medal."
Murray bows out having won 739 of his 1001 ATP Tour level matches, earning 46 singles titles across his career as well as spending 41 weeks as world number one.
But in true Murray fashion, he posted to X joking he never liked the sport he played for 18 years shortly after the defeat.
The two-time gold medallist had twice come from the jaws of defeat at the tournament, but there was to be no miracle this time as they were beaten 6-2 6-4 in the quarter-finals.
Murray and Evans endured a nightmare start to proceedings, conceding a double break to their American opponents which proved to be decisive in deciding the opening set.
Despite a better start in the second, Paul and Fritz would again stamp their authority on the encounter, racing into a 5-2 lead and to within a game of victory.
But as shown in the previous rounds, Murray and Evans showcased grit and determination to claw themselves to within a game of levelling the contest after saving one match point in the eighth game.
However, Fritz's strong serving would come to the fore as it had done throughout the contest on Court Suzanne Lenglen, making no mistake to wrap up the victory.
Paul and Fritz will face Matthew Ebden and John Peers for a place in the final, potentially setting up an all-American clash with Austin Krajicek and Rajeev Ram.
Data Debrief: Murray's farewell ends in disappointment
There was to be no comeback from Murray and Evans this time around, as tennis says goodbye to one of its greatest ever players.
But the encounter belonged to the Americans, whose performances on serve proved to be the difference at Roland-Garros.
Fritz and Paul would win 63% of their 40 first serve points, winning seven service games compared to the Brits' four.
Musetti, who was beaten by Novak Djokovic in the semi-finals, prevailed 6-4 1-6 6-3 on Saturday at Roland-Garros.
It marked a historic success for Italy, bringing up the nation's first tennis player to win a medal at the Olympics since Uberto de Morpugo in 1924, with those Games having also taken place in Paris.
While Auger-Aliassime leaves empty-handed from the singles, he did win bronze in the mixed doubles.
Prior to Musetti's triumph, Tommy Paul and Taylor Fritz won in straight sets to claim the bronze medal in the men's doubles, beating Tomas Machac and Adam Pavlasek.
After falling to defeat to eventual gold medallists Matthew Ebden and John Peers in the semi-finals, the American duo dusted themselves down to claim a 6-3 6-4 victory.
Paul and Fritz controlled the opening set, earning a double break in the sixth game and, despite losing the next two, held on to take the early advantage.
The Americans squandered three match points to seal the win, but confirmed their place on the podium in the 10th game, with Team USA taking home two medals from the doubles after Austin Krajicek and Rajeev Ram's silver earlier in the day.
It was not all doom and gloom for Czechia, though, with Machac and Katerina Siniakova having claimed gold in the mixed doubles on Friday.
Murray's glittering career came to an end on Thursday in the Olympics men’s double quarter-finals as he and Dan Evans were beaten in straight sets by American pair Tommy Paul and Taylor Fritz.
He announced before the Games that it would be the final tournament of his career, and he pulled off two superb comeback wins, saving multiple match points in both, to make it to the last eight.
Murray bows out as a two-time Olympic gold medallist with three grand slams. He won 739 games across his 1001 ATP Tour level matches, winning 46 singles titles.
Five of those came at Queen's Club, and shortly after his retirement was confirmed, the newly named Andy Murray Arena was announced.
"Andy is the greatest tennis player ever to come from this country and a giant of British sport," LTA chief executive Scott Lloyd said in a statement.
"His contribution to the game is immense and has bought us all so many moments of pride. He was relentless in his pursuit of excellence and had a single-minded determination to succeed.
"We are proud to name the arena at our tournament at the Queen's Club after him."
Meanwhile, Novak Djokovic has led the tributes to Murray, having played him 36 times over the course of their careers.
"Andy, we were all hoping that this message will never come, but here it is," Djokovic said in a message uploaded to the ATP Tour's X account.
"I just want to say congratulations for an amazing career, multiple grand slams, number one in the world, two Olympic gold medals, a Davis Cup. You've done it all. Amazing, really.
"Congratulations to you, to your family. We go back a long time. I think we were 11 when we met the first time on the court, and we had some incredible encounters, and rivalries.
"We played finals of every single grand slam. You brought the Wimbledon title to Britain after 77 years, unfortunately, against me in the final, but it was quite a special historic moment that I cherish and was grateful to be part of.
"You can be proud of your incredible resilience and fighting spirit, the inspiration that you give to all of us after all the hardships you went through - the surgery, the restrictions in your movement.
"You still kept going, you showed everyone there is no letting go and no giving up. Kudos to you and all the best to you and your family. You deserve the best possible farewell."
