The Nationalists also seized on the appointment of Tory right-winger Priti Patel as a Treasury Minister. They warned that her promotion was a sign a No vote would lead to massive reductions to Scotland's budget given she was in favour of "swingeing cuts" to the annual £30 billion block grant.
However, last night the two-pronged attack was denounced by Scottish Secretary Alistair Carmichael as disgraceful scaremongering by a Yes camp, which, he said, was becoming shriller as it became more desperate.
As the Prime Minister made sweeping changes to his ministerial frontline, removing some of the "male, pale and stale" ministers and replacing them with a tranche of younger, up-and-coming female colleagues, the First Minister seized on the appointment of the successor to William Hague at the Foreign Office.
Mr Salmond referred to remarks made last year by Mr Hammond, who said he would vote for a British exit from the EU unless key changes in the relationship were made but stressed it would be defeatist to leave the Brussels bloc without seeking reform.
The First Minister said: "Philip Hammond's promotion to Foreign Secretary has put one hand on the exit door leading the UK out of the European Union.
"Mr Hammond is the first person to hold this post who has openly said he is in favour of quitting the EU. As such, this is the clearest signal yet that, under David Cameron, the UK is on the fast-track out of Europe."
Mr Salmond claimed the clear risk for Scotland, if it did not vote Yes, was being dragged out of Europe against its wishes with hugely damaging consequences for jobs and investment.
"Only a Yes vote in September can secure Scotland's place in Europe, giving us a seat and a voice at the top table; Westminster is dancing to a Ukip tune and that tune is leading only one way."
In a separate attack, the Nationalists criticised the appointment of Ms Patel as the Exchequer Secretary given that she had been previously quoted as saying the debate on Scotland's future provided a good opportunity to cut the Scottish budget.
The SNP also seized on remarks by Inverness-born Stephen Crabb, the new Welsh Secretary, who in 2007 as a self-professed "devo-sceptic", said: "Together with uncontrolled immigration and relentless European integration, devolution has the potential to cause huge and permanent damage to our country."
Angus Robertson, the SNP leader at Westminster, described Mr Cameron's reshuffle as a "warning of Scotland's fate if we voted No in September to stay under the Westminster system". He said it was "proof beyond doubt Scotland would be singled out for more Westminster cuts in the event of a No vote".
But Mr Carmichael hit back, branding Mr Salmond's attack on the new Foreign Secretary as ridiculous, saying: "We have all three party leaders in the UK from Labour, the LibDems and Conservatives, who are committed to the principle of the UK staying within the EU."
The Secretary of State also dismissed the SNP attack against Ms Patel as "disgraceful scaremongering".
He said: "Government policy is set by the Prime Minister on this and he wrote to Alex Salmond some time ago, making it clear nothing was changing with the Barnett Formula and that's Government policy."
Among the Coalition changes, the biggest surprise was the demotion for Michael Gove, who was moved from the Education brief to become Chief Whip.
The Scot, who will lead the Tories' election charge and effectively become Minister for TV, said he was happy with the move as it would allow him to "shape the agenda".
Mr Cameron defended the move, saying: "I wanted one of my big-hitters, one of my real stars, one of my great political brains - someone who has done extraordinary things for education in this country - to do that job, to deliver the Government's programme and to help secure the future for our country."
One of the main themes of the frontline changes was the promotion of the 2010 intake of women, including the promotion of Nicky Morgan as Education Secretary and Liz Truss as Environment Secretary.
But it emerged the new Leader of the Lords, Baroness Stowell, who will no longer be a full-time member of Cabinet, was to be paid less than her male predecessor as there was a legal limit on the number of full Cabinet ministers and Mr Hague was retaining his status and salary despite moving to become Commons Leader.
However, last night it was made clear Tory HQ would make up the difference in Lady Stowell's pay.