The Panelbase survey, commissioned by Yes Scotland, found 42% of people intend to vote Yes on September 18, compared to 46% voting No, and 12% undecided. Excluding the undecided voters, support for Yes was 48% against 52% for No.
The poll suggests the No campaign's lead has narrowed since the last Panelbase poll was published at the end of July, when the figures for decided voters were 46% Yes and 54% No.
The prospect of Scottish NHS budget cuts within the Union put support for independence in the lead among both men and women and pushed support among female voters over the crucial 50% mark.
Respondents were asked: "Does the prospect of an increased role of the private sector in the NHS in England having an adverse effect on the Scottish budget which funds NHS Scotland make you likely or unlikely to vote for an independent Scotland in the referendum?"
Excluding don't knows, overall 57% said it made them likely to vote Yes and 43% unlikely.
Among women voters, it was 52% likely and 48% unlikely, while for men it was 61% quite or very likely and only 39% unlikely. A key battle ground as the campaign enters its final month is for women voters, who have so far proved less inclined to support a Yes vote than men.
Better Together chairman Alistair Darling said the biggest threat to the NHS was the austerity which independence would trigger.
He said: "The clear and present danger to the NHS in Scotland is Alex Salmond and Alex Salmond alone."
Despite health already being run by Holyrood, the future of the Scottish NHS has emerged as a key part of the referendum in recent weeks.
The Yes campaign has warned that privatisation and budget squeezes in the NHS south of the Border could see knock-on effects, possibly including cuts or even privatisation in the NHS in Scotland.
Raising the alarm again last week, Alex Salmond promised to embed the right to free healthcare in a written constitution under independence.
Unionist parties accused him of scaremongering and pointed out that overall health spending in England was rising, not falling, and said talk of threatened cuts was simply a cynical invention.
New figures issued by the Scottish Conservatives this weekend also suggested the SNP has no objection in principle to private healthcare, as the Scottish NHS had spent £400 million buying patient services from private health firms since the SNP took office.
The last month of the campaign looks set to boil down to a clash between "Vote No to save the pound" versus "Vote Yes to save the NHS".
Various polls suggest that healthcare is a higher priority for female voters, whom the Yes campaign need to attract, than it is for men. Yes strategists also believe the NHS has a more visceral appeal than currency.
Also worrying for the No camp, Panelbase found 37% support for a Yes vote among those who gave Labour their constituency vote in the 2011 Holyrood election, implying 230,000 people are rejecting their party's appeal for a No vote.
Panelbase also found a smaller gender gap than in previous polls, with male decided voters hitting 52% for Yes and 48% for No.
For women the figures were 43% Yes and 57% No. More than half of respondents (54%) agreed that Scotland could be a small successful country, against just 31% who disagreed.
Deputy First Minister Nicola Sturgeon said: "These are extremely encouraging findings, showing rising support for Yes and putting the referendum on a knife edge.
"We have a lot of work to do, but Yes only needs a 2% swing to move ahead, and I am confident we can and will achieve this in the month ahead.
"A Yes vote is our one opportunity to protect Scotland's NHS from the damaging impact of privatisation south of the Border, and it is clear this issue is galvanising support for Yes.
"Yes moves ahead among women as well as men on the NHS issue, and it is vital that we come together as a country to protect Scotland's NHS by voting Yes on 18 September."
She also said there was no intrinsic difference in the attitude of men and women to independence, and women were "just taking their own time" to back a Yes vote.
Sturgeon said women were less likely to be moved by "the big symbols of statehood … flags and embassies and all that stuff" and were more interested in quality-of-life issues.
An all-women Survation poll published last week found 34% of women would vote Yes and 50% No, or 40% Yes and 60% No among decided voters.
Darling yesterday launched a furious attack on the Yes side's "cynical" focus on the Scottish NHS, calling it "a new phase of desperation".
Speaking to Labour activists in Coatbridge, he said: "A campaign that started by branding their opponents as scaremongers for telling the truth has ended up as the purveyors of threats and doom laden predictions which owe more to desperation that to reality.
"Take their claims on the National Health Service, which I find quite objectionable.
"Health, like education, is entirely devolved to Scotland. The only people who could privatise the NHS right now are the SNP.
"Be in no doubt. Alex Salmond is not offering an end to austerity. Alex Salmond is offering austerity plus.
"The clear and present danger to the NHS in Scotland is Alex Salmond and Alex Salmond alone. What Alex Salmond is asking Scotland to vote for is austerity plus for decades to come.
"The truth is, if you believe in a properly funded NHS, free at the point of use, you have to vote for Scotland to remain in the UK."
Panelbase conducted the poll between August 12-15 among 1,026 people aged 16 or over.