Home Secretary Theresa May has launched the government's updated strategy for tackling terrorism by saying there needs to be better focus on preventing extremism at community levels.
She said a review identified serious failings with the existing policy - known as Prevent - set up in 2007.
Some of the 63m annual budget had reached the extremist organisations it should have been confronting, she said.
Shadow home secretary Yvette Cooper accused Ms May of "point-scoring".
Speaking in the House of Commons, Ms May said the strategy had "failed to tackle the extremist ideology that not only undermines the cohesion of our society, but also inspires would be terrorists to seek to bring death and destruction to our towns and cities".
"In trying to reach out to those at risk of radicalisation, funding sometimes even reached the very extremist organisations that Prevent should have been confronting.
"We will not make the same mistakes," she said.
The revised strategy will see 36m spent and 25 areas in England will be targeted as priorities.
They include Birmingham, Leicester, Luton, Manchester, Leeds and some London boroughs.
It will also withdraw support from extremist groups - even non-violent ones - and cut off funding to those opposed to what the government calls fundamental British values.
Ms May said the strategy would be targeted against "those forms of terrorism which pose the greatest risk to our national security, currently al-Qaeda and those they inspire".
"But Prevent must also recognise and tackle the insidious impact of non-violent extremism, which can create an atmosphere conducive to terrorism and can popularise views which terrorists exploit," she said.
Mrs May added that funding would be removed from organisations that "do not support the values of democracy, human rights, equality before the law, participation in society".
"If they don't accept these fundamental and universal values, then we will not work with them and we will not fund them," she said.
Prevent was originally launched after the 7 July bombings in 2005 to stop the growth of home-grown terrorism.
Previously, Mrs May has said that, as a result of the strategy's review of government support, about 20 of the organisations that received funding over the past three years would have their cash withdrawn.
On Monday, Mrs May accused universities of complacency in tackling Islamist extremism - a charge denied by the vice chancellors' body, Universities UK.
She told the Daily Telegraph: "I don't think they have been sufficiently willing to recognise what can be happening on their campuses and the radicalisation that can take place. I think there is more that universities can do."
In February, Prime Minister David Cameron said organisations seeking to present themselves as a gateway to the Muslim community were showered with public money despite doing little to combat extremism.