The Prime Minister said his success continued a golden summer of sport, while the First Minister described Murray as a "Scottish sporting legend".
Mr Cameron tweeted: "Delighted Andy Murray is continuing a golden summer of sport by winning the US Open. A truly great victory."
Mr Salmond, who congratulated Murray on behalf of his entire Cabinet who met yesterday, said: "Congratulations to Andy Murray on what was a fantastic performance. 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."
Scottish Secretary Michael Moore added that the 25-year-old was a "worthy champion" who has fought his way to the top of the game.
He said: "It was fantastic to see Andy Murray win his first Grand Slam. He has capped off a sensational summer of UK sport with his achievement and the whole of Scotland can be proud of him."
Johann Lamont, Scottish Labour Leader, said: "To win at the fifth attempt beats Robert the Bruce – by legend he lost six times before leading his army to victory. This is truly well deserved.
"Andy Murray has shown enormous physical and mental strength, never giving up and always believing in himself.
"Victory at the US Open rounds off a great summer of sport. Andy Murray has done Scotland and Britain proud this year. A grateful nation salutes him."
Murray beat Djokovic 7-6, 7-5, 2-6, 3-6, 6-2. The five-hour epic saw the sportsman dig deep to secure victory in the final set of the match, which has been described as "world class" by Tennis Scotland chief executive, David Marshall.
Mr Marshall added: "He always said he wanted to win slams and now he can aim for world number one.
"When you have an iconic role model like Andy it really makes it easier to profile the sport."
Mr Marshall suggested the postbox which was painted gold in Murray's hometown of Dunblane to mark his Olympic success become platinum following his Grand Slam win.
Louise Martin, sportscotland chairwoman, added: "Andy's determination and dedication to the sport is an inspiration for youngsters across Scotland, and it is fantastic that he has now added a Grand Slam win to his Olympic gold."
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