Business secretary Vince Cable announced the switch in policy following a review of the site blocking measures - a key component of the Digital Economy Act - by media regulator Ofcom.
Internet Service Providers such as BT and TalkTalk have objected to the proposal that copyright owners should be allowed to force them to cut off some websites.
Last week, the Motion Picture Association won a court injunction requiring BT to block access to Newzbin2, a site that allegedly hosts links to illegal copies of movies and music.
The action was taken under the Copyright, Design and Patents Act rather than the Digital Economy Act, which has led some observers to question the need for specific legislation.
Cable acknowledged that more must be done to protect the copyright holders, but nevertheless he said that the site blocking legislation in the DEA would be dropped.
Ofcom had warned the government that none of the various techniques for blocking websites was "100% effective", while each carried a cost, impacted on network performance and "ran the risk of over-blocking".
"All techniques can be circumvented to some degree by users and site owners who are willing to make the additional effort," said the regulator.
Speaking to BBC News, Cable appeared to indicate that the landmark Newzbin2 case had opened up other legal avenues for combating online piracy.
"We've discovered that the drafting of the original laws, which took place a year or so ago, were not tight," he said.
"There are test cases being fought in the courts, so we're looking at other ways of achieving the same objective, the blocking objective to protect intellectual property in those cases, but in a way that's legally sound."
The government acknowledged that the court process can be lengthy, meaning there would be limited options for tackling illegal websites that pop up for a short period of time and then shut down. But ministers said that they are exploring options for a fast-track legal process for site blocking to benefit rights holders.
Also today, Cable announced a raft of "big changes" designed to update the UK's copyright laws, based on the Hargreaves Review published in May.
The most significant development is that the government intends to legalise "format shifting", essentially the ripping of CDs and DVDs for personal use. Millions of people regularly convert movies and music to other formats, without realising that it is technically illegal.
Cable said the approch would bring the law "more up-to-date to have a proper balance which allows consumers and businesses to operate more freely, but at the same time protect genuinely creative artists and penalise pirates".
The minister claimed the economy would benefit to the tune of around £8bn over the next few years by updating the UK's copyright legislation.
However, he did not indicate how the new rules would apply to customers using the emerging range of cloud-based storage systems, such as those launched by Google, Amazon and Apple recently in the US.
Cable also plans to enact the Hargreaves Review recommendation to relax the rules on parodies and reworkings of existing content, such as recent viral internet video 'Newport State of Mind'.
The song, performed by a Welsh rap duo, parodied the Jay-Z and Alicia Keys hit 'Empire State of Mind' and became a hit on YouTube, but was taken down following a copyright claim by EMI.