The former prime minister said the six changes he was proposing would transform the "unitary and centralised" system in the UK.
As part of that, he suggested that the Scottish Parliament should raise about 40% of the cash it spends and said giving Holyrood more powers over income tax was the best way to achieve this.
Mr Brown has submitted his proposals to Labour's devolution commission, which will shortly put forward plans to boost devolution in the event of a No vote in September's Scottish independence referendum.
In a speech in Glasgow, the former Labour leader said: "We need to build the future of the relationship between Scotland, England and the rest of the United Kingdom.
"I believe there are six constitutional changes we have got to make for a better relationship between Scotland and the rest of the United Kingdom, to turn what I would call a unitary and centralised state of the past into a partnership of equals and one where there is power-sharing across the United Kingdom."
Mr Brown spoke out at the same time as former Liberal Democrat leader Sir Menzies Campbell said political parties should meet within 30 days of a No vote in the independence referendum to agree further powers for Scotland.
Sir Menzies said politicians should take advantage of the public appetite and momentum for more powers created by the independence referendum to act swiftly towards further devolution.
As part of his submission to Labour's devolution commission, Mr Brown argued more tax-raising powers should come north.
Scotland is already due to get new powers over income tax from April 2016, when the UK Treasury will deduct 10p from standard and upper rates of income tax in Scotland and give MSPs the power to decide how to raise cash.
Mr Brown went further than this, though, saying: "The first 5p of income tax should be decided by the UK Government, the next 15p by the Scottish Government.
"I believe that is a fair way of raising 40% of the revenue of the Scottish Parliament in Scotland."
This, he argued, would make Holyrood ministers "accountable to the people of Scotland for the way that money is spent", adding: "They will have to answer at elections, which they don't have to do at the moment, for tax decisions that they make."
He also appeared to indicate the Scottish Parliament could get some power to alter the top rate of income tax, saying that "if the fairness of the Scottish Parliament was being undermined by unfairness elsewhere then there should be power to do something about it".
Mr Brown rejected devolving corporation tax - something the Scottish Government has called for - saying this risked creating a "race to the bottom".
He also said European laws prevented Scotland being given control over VAT and argued National Insurance should not be devolved as it pays for pensions and welfare across the UK.
As well as devolving income tax powers, Mr Brown called for a new division of powers between Scotland and Westminster that gives Holyrood more responsibility in areas such as employment and transport.
At the same time, he said more powers should be handed down to a local level.
He called for a new UK constitutional law, backed by an historic document which he said would be equivalent to a bill of rights, which would set out the purpose of the UK as being to pool and share resources for the defence, security and well-being of the citizens of all four nations.
He added there should be a constitutional guarantee of the permanence of the Scottish Parliament, with this to include provisions which would prevent Holyrood from being either undermined or over-ruled.
Politicians should also be required to work together in partnership arrangements to end the "permanent stand-off between a Scottish Government and the rest of the UK", Mr Brown said.
He stressed: "We want to separate Scotland not from the rest of the United Kingdom but from poverty.
"We want people to have the freedom from deprivation and homelessness, not the freedom from our neighbours in other parts of the country.
"The liberation struggle we are fighting for is for every person to have the chance to make the most of their potential in life.
"We see why it makes sense for Scotland with its own parliament to work with the rest of the United Kingdom, and indeed Europe and the rest of the world, to achieve common ends".
He highlighted the UK's history of "pooling and sharing resources", saying this had developed "so that those areas and that those people in greatest need would get the best help".
He added: "I believe it is in the people of Scotland's interest to maintain this system of pooling and sharing resources across the United Kingdom."
Mr Brown argued it should be made "explicit so that nobody has any doubt that the purpose of the United Kingdom is to pool and share out resources, for our common defence, our security, but most of all so that every citizen, whether Scottish, English, Welsh or Irish, has the same rights, the same economic rights, the same social rights and the same opportunities, as anybody else".
Guaranteeing the permanence of the Scottish Parliament would "end the constitutional fiction that somehow there is undivided Westminster sovereignty", he said.
This would be a "major statement of constitutional purpose" that "recognises that the future is based on a partnership of equals".
He also said it was time for a fresh look at the division of powers between the UK and Scottish parliaments.
"I am clear there are powers of the Scottish Parliament that could be enhanced and would enable it to be closer to the people of the country in making decisions," Mr Brown said.
"I also say there should be a transfer of powers from the Scottish Parliament and Westminster to local communities.
"We should support more powers over economic regeneration to be held in Glasgow here and in other parts of the country, and the work programme and help for the unemployed could be given better in many cases by local authorities than by central government.
"So, we need to think very closely and clearly about a radical division of powers that makes it possible for us to devolve decisions to those areas where decisions can best be made."
He also stressed the need for politicians from Holyrood and Westminster to work together to tackle issues such as poverty, unemployment, housing and the environment, arguing these could only be addressed by the two governments working together.
Mr Brown said: "If we have a new constitution which is based on risk sharing, resource sharing, power sharing, if there is a partnership between the nations of the United Kingdom in the future, we've got to find a way of working together again.
"We cannot have this permanent stand-off between a Scottish Government and the rest of the UK, a war of attrition that is being fought with no co-operation, even when there are issues where there is a duty for all of us to work together.
"We need to provide in the legislation in future a requirement, even for those people who don't like each other, to have to talk about the problems that cannot be solved unless people work together to solve them."