The TNS survey found that 60 per cent of people did not know that nearly half of their income tax will be set by the Scottish Parliament from 2016, as a result of laws passed in 2012.
The poll came as both sides in the debate continued to clash yesterday over pledges to devolve even more power to Holyrood in the event of a No vote.
Labour, the Conservatives and LibDems this week signed a joint statement highlighting their commitment to further devolution, including greater control over income tax, if voters in Scotland reject independence in September's referendum.
But the SNP, citing a Panelbase poll last week which showed barely one-third of Scots believed more powers would follow a No vote, said voters "will not be fooled" into accepting their promises.
The TNS survey of just over 1000 voters found 60 per cent were unaware that powers were about to be transferred to Holyrood under the 2012 Scotland Act.
A further 39 per cent were aware, while others were unsure.
The 2012 Scotland Act followed the Calman Commission, backed by the three main pro-UK parties, which concluded the Scottish Parliament should be more responsible for raising the money it spends.
Among a range of new powers devolved under the act, Holyrood will replace stamp duty with its own tax on property transactions from next year. An even bigger change will take place in 2016, when Holyrood will become responsible for setting nearly half the income tax paid by Scots.
TNS found a significant appetite for more powers to be devolved. One-third of voters backed the transfer of more powers over tax and welfare, making that a more popular option than independence (28 per cent) or the status quo (26 per cent).
Professor Jim Gallagher, an adviser to Better Together, said the 2012 Scotland Act had fallen victim to a "conspiracy of silence".
He said: "It suits the nationalists to claim the UK is completely inflexible. They'll never talk about the Scotland Act."
Stephen Noon, chief strategist at Yes Scotland, said the No parties admitted the new Scotland Act does not go far enough. He added: "The No parties all propose very minor additions to the new powers that are already planned."
A separate survey published yesterday found 41 per cent wanted Holyrood to have more powers but felt they would be delivered without independence.
The poll, for the British Election Study, was conducted as part of the Economic and Social Research Council's Future of the UK and Scotland project.