The 19-year-old dragged his weary body to a 6-4, 2-6, 7-6 (7/1), 6-3 victory over Norway's Casper Ruud in the final.
Alcaraz, the first teenager to claim the top ranking, is the youngest Grand Slam men's champion since Rafael Nadal at the 2005 French Open after a performance which yielded 55 winners and 14 aces.
On a day of landmarks, he is also the youngest champion in New York since Pete Sampras in 1990.
"This is something that I dreamt of since I was a kid -- to be number one in the world, to be a champion of a Grand Slam," said Alcaraz.
"This is something that I worked really really hard for. It's tough to talk right now, a lot of emotions.
"This is something that I tried to achieve, all the hard work that I did with my team and my family. I'm just 19 and all the top decisions were taken by my parents and my team. This is something that is really, really special for me."
It was a gruelling tournament for Alcaraz who also claimed the record for most time spent on the court at a single Grand Slam event, passing the 23 hours and 21 minutes it took Kevin Anderson to finish runner-up at Wimbledon in 2018.
Most of his 23 hours and 40 minutes were taken up by the more than 13 hours it took him to play three successive five-setters to reach the final.
He had to save a match point in his five-hour 15-minute quarter-final win over Jannik Sinner.
Nadal was quick to congratulate the player widely seen as his heir apparent.
"Congratulations @carlosalcaraz for your first Grand Slam and for number 1, which is the culmination of your great season, which I am sure will be many more," tweeted the 22-time Slam title winner.
Defeat for Ruud, who was also vying for the world number one ranking, was his second in a Slam final this year after he was routed by Nadal in the French Open.
"Today was a special evening for both Carlos and I. We knew what was at stake, and I think it's fitting both finalists will be number one and number two in the world," said Ruud.
"I'm happy with that number and I will continue to chase that ranking and my first Grand Slam."
With the roof closed, the Arthur Ashe Stadium crowd observed a moving moment's silence on the 21st anniversary of the September 11 attacks before the final got off to a tentative start.