continuing Britain's "golden summer of sport" following the Olympics and Paralympics.
British tennis insiders meanwhile praised the influence of coach Ivan Lendl in turning Murray into a Grand Slam champion.
"Delighted Andy Murray is continuing a golden summer of sport by winning the US Open. A truly great victory," Cameron said on Twitter.
Scottish First Minister Alex Salmond said Murray was now a "legend" in his homeland.
"Congratulations to Andy Murray on what was a fantastic performance," he said in a statement.
"This is another brilliant win over Novak Djokovic and continues an amazing year for Andy. Now Olympic and US Open champion, Andy truly is a Scottish sporting legend and I'm certain that more grand slam titles will follow."
Roger Draper, chief executive of the Lawn Tennis Association, the sport's governing body in Britain, said Murray winning the Olympic men's singles gold medal last month and following it up with the US Open title was "a phenomenal achievement," he told BBC radio.
"He's done it in an era where you've got not just Roger Federer -- the greatest player that's probably ever lived -- but you've got Rafa Nadal, Novak Djokovic as well."
Former British number one Greg Rusedski, who lost in the 1997 US Open final, said Lendl had a big hand in Murray's victory.
The Scot "found a way to get it done and found a way to control his emotions.
"He can thank Ivan Lendl for that. You have to give him so much credit for what he's done, to keep believing in what he's done," he told Sky Sports television.
"It shows you what a champion he is and, having won this, he can go on to win many majors and maybe end the year as the world number one."
Former British number one Roger Taylor, 70, a four-time Grand Slam semi-finalist, added: "So much confidence has come from Andy's Olympics win and Lendl has added a great presence.
"Andy is on top of the world, very few people can beat him and now with the added confidence from the Olympics I am sure he will go on to win many grand slam titles and become number one in the world.
"Finally British tennis is back on the map."