The Italian top seed wasted three match points at 5-3, 40-0 up in the fourth set and was also 6-5 up in the decider, before Alcaraz clawed his way back in dramatic fashion to win the title for the second year running.
It was the longest French Open final since tennis turned professional in 1968, lasting five hours and 29 minutes.
"I won't sleep very well tonight, but it's okay," Sinner said during the presentation ceremony.
"First of all, Carlos, congrats. An amazing performance, amazing battle, amazing job. I am very happy for you, you deserve it.
"It's easier to play than to talk now," said the 23-year-old, who was on a 20-match winning streak in the majors.
Sinner had only played one tournament, in Rome in May, after coming back from a three-month doping ban, but said his run to the final proved he was on the right track.
"I was ready. I was feeling much more ready than in Rome. I think we saw that. He's the world number two, but he's number one on clay," he told a press conference before listing his missed opportunities in the match.
"I was a break up in the third, was a break up in the fourth, had three match points, serving for the match," he said.
"I came back. Six-five up (in the fifth set), I had chances also in the fifth. So many chances I couldn't use."
The two finalists, who have won seven of the last eight Grand Slams to stamp their authority on the tour, were locked in a fierce battle that mesmerised the 15,000 crowd at the Philippe Chatrier court.
"When it was over, it was over," Sinner said. "That's a different feeling, different things coming through your mind."
"You cannot change anymore when the match is over. But when you start a fifth set, you can still change some things."
It was the first major final between two men born in the 2000s, while Alcaraz became only the second man in the professional era to win all of his first five Grand Slam singles finals after Roger Federer.
"I had lots of chances, but this is the good part of the sport. Also, today it got me the sad part, no?," Sinner said. "But, you know, if you watch only the sad part, you're never going to come back."
