"It is what I have been working towards for the last 10 years of my life," Murray told BBC Sport in New York. "It means the world to me."
Tom FordyceChief sports writer, BBC Sport
"Andy Murray's nerve-mangling, history-making US Open triumph over Novak Djokovic was many things: one of the great finals of the modern era, a late-night thriller from the city that never sleeps, a breath-taking demonstration of physical strength and mental fortitude. It was also the perfect bookend to a few months that British sport can scarcely believe and will never forget."
The last British man to win a Grand Slam title was Fred Perry, who won the US Open in 1936, 76 years ago.
Murray has long been tipped to emulate Perry, who won eight majors, but had lost in his four previous Grand Slam finals.
He finally ended his barren run at the fifth attempt, beating defending champion and world number two Djokovic 7-6 (12-10) 7-5 2-6 3-6 6-2 in a contest at Flushing Meadows that lasted almost five hours.
"I think everyone is in a little bit of shock that it has happened," said Murray, speaking to the media after his win. "I've seen my mum after I've lost Slam finals and she has been really upset. Everyone's really happy.
"It is hard to explain. It has been a long, long journey to this point, so I don't know if it is disbelief or whatever, but I am very, very happy on the inside. I'm sorry if I'm not showing it as you would like."
Asked what it felt like to end Britain's lengthy wait for a male Grand Slam winner, Murray replied: "I've been reminded of that most days of my life for the last few years. It's great to have finally done it and I don't need to get asked that anymore."
"To come back in the next Grand Slam and to win it in a five-set match against Djokovic on a hard court - he hasn't lost for a couple of years on a hard court in a Grand Slam - is unbelievable," said Murray.
Sir Alex Ferguson on Andy Murray's US Open win
"I'm really proud for the boy. That was a real test of a champion for me. I love watching tennis. It was more nerve-wracking than a Premier League match. I'm usually in control of my own situation but I wasn't in control tonight."
He said it would have been hard to recover had he lost a fifth major final.
"I'm very, very happy to come through," he said. "If I had lost this one from two sets up, it would have been tough to take."
Asked what had been the key to success at Flushing Meadows, Murray said: "I have kept improving.
"I have worked hard all the time, even after the tough losses. I have had a fairly solid team around me as well for a long time."
He also credited coach Ivan Lendl after teaming up with the former world number one and eight-time Grand Slam winner in January.
Murray can win more slams - Henman
"Having Ivan Lendl around has definitely helped, not just me but the rest of the team as well, having someone of his experience, especially in these situations," said Murray.
"He's got his name on that trophy three times and made the final eight times in a row. It really helps having him around."
Murray said he also hoped his win "inspires some kids to play tennis" and "takes away the notion that British tennis players choke or don't win".
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.