The 19-year-old Spaniard defeated Norway's Casper Ruud 6-4, 2-6, 7-6 (7/1), 6-3 in the final to become the youngest Grand Slam men's champion since Rafael Nadal at the 2005 French Open.
"Right now I'm enjoying the moment. I'm enjoying have the trophy in my hands but of course, I'm hungry for more," said Alcaraz, the first teenager to be number one in the rankings.
"I want to be in the top for many, many weeks and I hope many years. I'm going to work hard again after this week, these amazing two weeks. I'm going to fight for have more of this."
Sunday's victory comes in a breakthrough year for Alcaraz.
His US Open title followed Masters wins in Miami and Madrid -- where he defeated Nadal and Novak Djokovic back-to-back -- Rio and Barcelona.
"Since I won Miami, I thought I was able to have a Grand Slam in my hands. But before Miami, I was thinking that I have to still grow up," he said.
However, that optimism faded with a last-16 exit at Wimbledon, a first-up loss at the Montreal Masters and a quarter-final defeat in Cincinnati.
"In Montreal and in Cincinnati I lost the joy a little bit. I felt the pressure. I couldn't smile on court which I'm doing in every match, every tournament," he explained.
"I came here just to enjoy, to smile on court, to enjoy playing tennis. If I have fun, I saw my best level, my best tennis."
Alcaraz's coach Juan Carlos Ferrero said the champion was "born to play these tournaments" and that there was much more to come from his compatriot.
"I think he's on 60% of his game. He can improve a lot of things. He knows and I know that we have to keep working," said Ferrero.
"Once to get to the No. 1, it's not done and you go. You have to keep working, keep playing at a huge level on that tournaments to keep winning.
"He knows that and I know that. I'm going to be very close from him to remember him."
Ferrero was French Open champion in 2003, the year that Alcaraz was born.
He has been guiding the new US Open champion since he was 15 when he was nowhere near the finished product.
"When he arrived to the academy when he was 15, he was like a spaghetti, very thin. We had to work," recalled Ferrero.
"Obviously we saw that he had very fast hands, very fast legs, but no muscles at all, not in the back, not in the legs.
"We had to work a lot. But obviously we saw something very special on him."
Alcaraz, however, was quick to point out on Sunday that he still only has one Slam title -- 21 behind Nadal's men's record.
"But I'm on the road," he admitted.
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