An apparent 'denial of service' attack, made it impossible to access the Home Office website for at least an hour.
Those trying to access the website were instead confronted by a notice that "Due to a high volume of traffic this page is currently unavailable."
The attack appeared to have been partly in protest at extradition proceedings against Gary McKinnon, 46, who is accused of hacking US military computers.
Other posts about "draconian surveillance proposals" suggested the hackers were also angry about recent Government draft proposals that would potentially allow the security services to monitor every email, phone call and website visit to see who people were contacting and what websites they were looking at.
On Twitter, messages purporting to be from the hacking group, were posted under the name AnonOpUK, saying: "Anonymous is famous …. UK Home Office. Maybe you should start to listen to the people."
Another message, apparently urging the hackers to continue the attack said: "Keep firing!"
A Home Office spokesman said: "We are aware of some reports that the Home Office website may be the subject of an online protest. We have put all potential measures in place and will be monitoring the situation very closely."
He said he was not in a position to discuss who might be mounting the suspected attack or why.
The Home Office added that if a successful denial of service attempt did occur tonight, it would "liaise with the technical team and update as necessary".
A denial of service attack prevents a website from functioning properly, sometimes by swamping it with more traffic than it can handle.
Such an action was believed to have been responsible for crashing the Home Office site.
The apparent attack came after it emerged last week that the Government was planning a massive expansion of its powers to monitor the email exchanges and website visits of every person in the UK.
Under legislation expected in next month's Queen's Speech, internet companies will be instructed to install hardware enabling GCHQ - the Government's electronic "listening" agency - to examine "on demand" any phone call made, text message and email sent, and website accessed, in "real time" without a warrant.
Ministers have faced a backlash over the plans, with senior MPs from both coalition parties, as well as civil liberties groups, lining up to denounce it.
The move has been condemned by opponents as an unnecessary extension of the state's powers to "snoop" on its citizens.
Anonymous, whose genesis can be traced back to a popular US image messaging board, has become increasingly politicised amid a global clampdown on music piracy and the international controversy over the whistleblowing website site WikiLeaks, with which many of its supporters identify.
Authorities in Europe, North America and elsewhere have made dozens of arrests, and Anonymous has increasingly attacked law enforcement, military and intelligence-linked targets in retaliation.
One of Anonymous's most spectacular coups was secretly recording a conference call between US and British cyber-investigators tasked with bringing the group to justice.
The collective has no real membership structure, with hackers, activists, and supporters able to claim allegiance to its freewheeling principles at their convenience.