The survey from pollsters YouGov showed that 40% of people quizzed wanted the UK to cut ties with Brussels.
Only 38% in the internet poll of more than 11,000 voters wanted the country to remain a member of the bloc.
It was the first YouGov poll to show the campaign to quit the EU in the lead since November last year.
In June, the "stay" side of the debate had a 10-point lead.
The poll also found that a majority of Tory voters (51%) and nearly a third of potential Labour voters (31%) wanted Britain to quit the EU.
Euro-sceptics seized on the poll last night as confirmation that the momentum in the campaign is swinging towards the exit camp.
PM David Cameron in front of Europe signGETTY
David Cameron is facing mounting pressure from the British public over his pro-EU stance
Jayne Adye, director of the cross-party anti-Brussels campaign Get Britain Out, said: "The YouGov poll has revealed the great British public is now seriously considering the full implications of our membership of the EU.
The result shows that we are winning arguments and we are winning by all working together
Douglas Carswell, Ukip MP
"The ongoing migration crisis has revealed the dominance of Germany as the driver of EU policy, willing to ride roughshod over the wishes of nation states.
"Whilst the OUT campaign is driven by the desire to see Britain prosper in an increasingly global world, the IN side remain dependent on scaremongering and peddling myths from Brussels.
"The British public realise the only way to secure our country's future is to Get Britain Out."
UK Independence Party MP Douglas Carswell said: "This is the best news for Euro-sceptics for a long time.
"It shows that we are winning arguments and we are winning by all working together.
"But we must not be complacent. Peter Mandelson, Tony Blair and the big corporate interests they represent are going to be doing everything they can to try and keep the Brussels racket going.
Read more:
http://report24uk.info/article/319565/support-for-crusade-to-get-uk-out-of-eu-surges-as-nearly-half-want-britain-to-leave-unionWith the refugee crisis and the Calais camps in the news nightly, it's no huge surprise.
