Mr Salmond called for greater tax-raising powers for the Scottish Parliament, after Mr Brown said the present system posed a "problem".
The Prime Minister said that, under the present system, the Scottish Parliament was accountable for the way it spends its cash - but not for the size of its budget, which comes from Westminster and is not linked to the success of the Scottish economy.
Mr Brown's comments, to the Scottish CBI in Glasgow, were taken as hinting at the possibility of more tax powers for Holyrood.
The issue could also be addressed, however, by "assigned revenues" - some taxes raised in Scotland staying in Scotland, without any need for new powers for Holyrood.
If that is what Mr Brown has in mind, Mr Salmond insisted it was not enough.
He told a financial conference in Aberdeen: "As we've seen just this week, there's growing support for an increase in the powers and responsibilities of the Scottish Parliament.
"The more economic and financial responsibilities we have, the better Scotland can do.
"It is essential that we have the ability to grow the economy through greater fiscal autonomy, not just assigned revenues."
Earlier, Scotland's Finance Secretary said Mr Brown had "caved in" to the case for more powers.
Growing support for the SNP and Labour's defeat in the Glasgow East by-election had forced Mr Brown to shift ground, said John Swinney.
In his speech to the Scottish CBI, Mr Brown launched a fierce attack on the SNP, accusing the Nationalists of a "bleak separatist obsession" and attacking their claim to be a pro-business party.
An official commission headed by Sir Kenneth Calman is studying whether the Scottish Parliament should have more powers, although independence has been excluded from its remit.
Mr Brown told the Scottish CBI: "I am not going to pre-judge the commission's work but I do want to say two things about it.
"First of all, devolution has worked, but I do see one problem - while there have been good reasons why this is so, the Scottish Parliament is wholly accountable for the budget it spends but not for the size of its budget.
"And that budget is not linked to the success of the Scottish economy.
"That is why we asked the Calman Commission to look carefully at the financial accountability of the Scottish Parliament.
"And this is a critical part of Calman's remit."
Mr Brown went on: "The second thing is more important still.
"Be under no illusion about my purpose.
"Devolution is intended to preserve the unity of the United Kingdom - and developing devolution is intended to strengthen Scotland's place within it."
Mr Swinney said: "Gordon Brown has caved in to pressure from the Scottish National Party and rising support for the SNP, based on the aspirations of the people of this country."
Mr Swinney also recalled: "People said that devolution would kill the SNP stone-dead - and we are now the Government of Scotland."
The Prime Minister today visited Glenrothes but Labour insisted it was not a campaign visit and that he would not be accompanied by Lindsay Roy, Labour's candidate.
No date has yet been set for the Glenrothes by-election, in which Labour is defending a majority of 10,664 at a time when it is recovering from the shock of losing Glasgow East to the SNP.
Mr Brown opened a trade union learning centre in Glenrothes and visited a voluntary centre in Levenmouth.
Scottish Liberal Democrat leader Tavish Scott challenged Mr Brown to back Lib Dem calls for Scotland to raise the money it spends.
"The future of Scotland is too important for political games," Mr Scott said.
"If he means it, I welcome him to the debate. The test will be Labour's words and deeds in the Calman Commission over the next weeks and months."
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