The Panelbase poll, commissioned by the SNP, put support for a Yes vote at 44%, compared with 43% for a No vote.
The result counted those certain and highly likely to vote in the referendum, out of a sample of 1043 Scots aged over 18.
It was strikingly different from the findings of a YouGov survey at the weekend which showed 29% in favour of independence compared with 59% opposed, one of the biggest margins recorded.
The latest Panelbase poll is the first to show the Yes camp in the lead since the question which will be asked in the referendum was finalised last year.
The last poll to indicate support for independence was ahead was a TNS poll in The Herald in August 2011, which put Yes at 39% and No at 38%.
Support for a Yes vote has increased by seven points since the last Panelbase poll in July, while backing for a No vote has fallen by three points.
SNP Depute Leader and Deputy First Minister Nicola Sturgeon said: "It is game on for next September's referendum. With just over a year to go, this poll indicates that the positive case for Yes is capturing people's imagination, while the No campaign's 'Project Fear' is running out of steam.
"The poll also shows that a clear majority of people back the Yes campaign's contention that Scotland could be a successful independent country - a common-sense proposition which the No campaign daily denigrate. And they trust the Scottish Government rather than Westminster to take decisions for Scotland by a margin of nearly four-to-one. These findings all underline the strength and potential of the Yes vote.
"Polls will move up and down between now and next September as the debate over Scotland's future gathers pace, but these very encouraging figures show that Yes can win and I believe will win. Certainly, the No campaign can no longer take the people of Scotland for granted."
The Yes camp's lead is based on the views of 908 voters who rated themselves eight out of 10 or higher in terms of how likely they were to vote (with 10 being certain to vote).
The poll showed that 93% of Yes supporters are very likely to vote in the referendum, compared to 88% of pro-UK supporters.
Of the full sample of 1043, support for staying in the UK retained a narrow lead, with 42% backing a No vote compared with 41% supporting a Yes vote.
A detailed breakdown showed that 24% of people who voted Labour in the Scottish Parliament constituency vote in 2011 intend to vote Yes.
Among women aged 35-54, the Yes camp lead the No side by 45% to 41%.
The poll also found 60% trusted Holyrood to take decisions for Scotland, compared to just 16% who trusted Westminster.
Ms Sturgeon added: "As we move closer to the referendum voters are beginning to make up their minds - and this poll suggests that as undecided voters find out more about the opportunities of independence, they are backing a Yes vote in increasing numbers.
"Scotland can be a successful independent country, and it is better for all of us if decisions about Scotland are taken by the people who care most about Scotland - the people who live and work here. That is the essence of the case for Yes, and it is a compelling and positive case that the No campaign has no answer to."